Khinchin Families and Hayman Class

Alicia Cantón; José L. Fernández; Pablo Fernández; Víctor J. Maciá

doi:10.1007/s40315-021-00420-6

Khinchin Families and Hayman Class

Cantón, Alicia; Fernández, José L.; Fernández, Pablo; Maciá, Víctor J. 2021-12-01 00:00:00 We give criteria, following Hayman and Báez-Duarte, for non-vanishing functions with non-negative coefﬁcients to be Gaussian and strongly Gaussian. We use these criteria to show in a simple and uniﬁed manner asymptotics for a number of combinatorial objects, and, particularly, for a variety of partition questions like Ingham’s theorem on partitions with parts in an arithmetic sequence, or Wright’s theorem on plane partitions and, of course, Hardy–Ramanujan’s partition theorem. Keywords Khinchin families · Hayman admissible functions · Asymptotic formulae · Partitions · Analytic combinatorics Mathematics Subject Classiﬁcation Primary 30B10; Secondary 05A16 · 11P82 · 60F99 1 Introduction Walter Hayman, in A generalisation of Stirling’s formula,[16], introduced what are now called Hayman (admissible) functions, a concept which has become a cornerstone of asymptotic methods in Analytic Combinatorics. The notion of Hayman function may at ﬁrst glance appear as too speciﬁc and tailor-made for saddle point approximation and for the Laplace method to be applicable. But the Hayman class is a rich collection To Walter K. Hayman, in memoriam Communicated by James K. Langley. Research of J. L. Fernández and P. Fernández supported by Fundación Akusmatika. Research of Victor Maciá partially supported by grant MTM201457769 of Ministerio de Economía y Competitividad of Spain. B José L. Fernández joseluis.fernandez@uam.es Extended author information available on the last page of the article 123 852 A. Cantón et al. of functions, closed under certain operations (exponentials of Hayman functions are Hayman) and small perturbations (the product of a polynomial times a function in the Hayman class returns a function in the Hayman class); its usefulness in Analytic Combinatorics derives mostly from facts like those. A few years later, following an idea of Khinchin, Paul Rosenbloom [25]intro- duced in Complex Analysis a family of probability distributions, which we term here a Khinchin family, associated to any given power series with non-negative coefﬁcients. In this probabilistic framework, he interpreted Hayman functions as power series with Gaussian Khinchin families (see Deﬁnition 3.1) and as a show of its fruitfulness, Rosenbloom proved the Wiman–Valiron theorem on the maximal term of entire func- tions, essentially by applying Chebyshev’s inequality to the corresponding Khinchin family. Luis Báez-Duarte [2] added to this circle of ideas the notion of strongly Gaussian power series (see Deﬁnition 3.4) and a basic substitution theorem (see Theorem C in Sect. 3.2.1) quite useful for asymptotic purposes. Products of power series become, in the Khinchin family side, sums of independent random variables, and by appealing to Lyapunov’s approach to the central limit theorem, Báez-Duarte was able to show j −1 that the function (1 − z ) is strongly Gaussian, and to deduce the classical j =1 Hardy–Ramanujan partition theorem. See also Candelpergher–Miniconi, [6]. In this paper we follow this thread of Hayman, Rosenbloom and Báez-Duarte. We take full advantage of the notion of strong gaussianity and go back to Hayman’s basic function theoretical ideas to extract a direct and purely function theoretical criterion (Theorem 4.1) for belonging to the Hayman class—and so, being strongly Gaussian—for power series which do not vanish in their disk of convergence. We use this criterion, combined with Hayman’s asymptotic formula of Theorem B, to derive in a simple and uniﬁed manner asymptotics for the enumeration of an assort- ment of combinatorial objects. Further, we use Theorem 4.1 to show, with no intervention of the central limit theorem or the circle method, that generating functions of partitions, standard or with a variety of restrictions on parts, are in the Hayman class, and then with the additional and crucial contribution of Theorem C we derive asymptotic formulas for partitions with parts in an arithmetic sequence (Ingham’s theorem), plane partitions (Wright’s theorem), or colored partitions and, of course, Hardy–Ramanujan’s partition theorem. We should emphasize that the general approach of Khinchin families and Hayman class functions aims to give ﬁrst order asymptotic formulae, but not full asymptotic expansions. In forthcoming work, we plan to extend this approach of Khinchin families and apply it to further asymptotic estimations of coefﬁcients like those of inﬁnite products, of large powers of power series, of generating functions of a variety of trees, etc. As it turns out, [16] is one of the most cited papers of Hayman and, as far as we know, he did not pursue its topic any further. The present paper is a natural continuation of [16] once you have at your disposal the Khinchin families framework and the contributions of Báez-Duarte [2]. This paper is dedicated to the memory of Walter Hayman. Maybe, he would have been pleased with the extra range of direct applications, which the present paper contains, of his A generalisation of Stirling’s formula,[16]. 123 Khinchin Families and Hayman Class 853 Plan of the paper. Section 2 expounds the basic theory of Khinchin families. In Sect. 3, the notions of Gaussian and strongly Gaussian power series are introduced, their probabilistic and asymptotic implications, like Hayman’s asymptotic formula, Theorem B, are analyzed, and the Hayman class of functions is presented. In this Sect. 3 we present basic criteria for non-vanishing functions to be Gaussian, Theorem 3.2, and to be strongly Gaussian, Theorem 4.1. This last theorem is applied in Sect. 4 to verify that a number of exponential functions, mostly of combinatorial interest, are strongly Gaussian and to obtain asymptotic formulae for their coefﬁcients. Theorem 4.1 is also the basic tool for the analysis of asymptotic formulae for the number of partitions of various kinds which, with the help of Theorem C, is carried out in Sect. 6. Notations. For positive sequences (a ), (b ), the notation a ∼ b as n →∞ means n n n n that lim a /b = 1; and we refer to that as an asymptotic formula. With a b n→∞ n n n n we abbreviate that c ≤ a /b ≤ C for positive constants c, C. Analogous notations n n are used for positive functions. denotes the distribution function of the standard normal. With ‘ogf’ and ‘egf’ we abbreviate, respectively, ordinary and exponential generating function. The disk of center z ∈ C and radius r > 0 is denoted D(z, r ). The unit disk D(0, 1) is denoted simply by D. E( Z ) and V( Z ) are reserved for the expectation (mean) and variance of the random variable Z.If X and Y are random variables, with X = Y we signify that X and Y have the same distribution. We write {x } for the fractional part of the real number x. The Bernoulli numbers are denoted by B , for j ≥ 0 (note that B =−1/2), while B (x ),for j ≥ 0, stands for the Bernoulli 1 j polynomials. For f ∈ C [0, N ], with k ≥ 1, the Euler summation formula of order k reads: f ( j ) = f (t ) dt j =0 k−1 1 1 B j +1 j +1 ( j ) ( j ) + f (0) + f ( N ) + (−1) f ( N ) − f (0) 2 2 ( j + 1)! j =1 B ({t }) k+1 (k) + (−1) f (t ) dt . (1.1) k! We will frequently refer to the comprehensive treatise Analytic Combinatorics [9] by Flajolet and Sedgewick for background on combinatorial issues. 2 Khinchin Families We denote by K the class of non-constant power series f (z) = a z , n=0 123 854 A. Cantón et al. with positive radius of convergence, which have non-negative Taylor coefﬁcients and such that a > 0. These are the power series of interest in this paper. Observe that f (t ) is increasing for t ∈[0, R), and, since f (0)> 0, we have f (t)> 0, for t ∈[0, R). Deﬁnition 2.1 The Khinchin family of a power series f in K with radius of convergence R > 0 is the family of random variables ( X ) with values in {0, 1,...} and with t ∈[0, R) mass functions given by a t P( X = n) = , for each n ≥ 0 and t ∈ (0, R), f (t ) while X ≡ 0. Any Khinchin family is continuous in distribution in [0, R). No hypothesis upon joint distribution of the variables X is considered: families, not processes. 2.1 Basic Properties 2.1.1 Mean and Variance Functions For the mean and the variance of X we reserve the notations m(t ) = E( X ) and t t σ (t ) = V( X ),for t ∈[0, R). In terms of f , the mean and variance of X may be t t written as tf (t ) m(t ) = ,σ (t ) = tm (t), for t ∈[0, R). (2.1) f (t ) For each t ∈ (0, R), the variable X is not a constant, and so σ (t)> 0. Conse- quently, m(t ) is strictly increasing in [0, R), though, in general, σ(t ) is not increasing, like in the case of polynomials. We denote M = lim m(t). (2.2) t ↑ R 2.1.2 The Case M =∞ This case M =∞, quite relevant in what follows, holds except in some exceptional cases, which we now specify. Let f ∈ K with radius of convergence R > 0 and assume that M < ∞. Then na t tf (t ) n=0 () m(t ) = = ≤ M , for t ∈ (0, R). a t f (t ) n=0 For any ﬁxed t ∈ (0, R), we then have upon integration in () that f (t ) t () ln ≤ M ln , for t ∈[t , R). f 0 f (t ) t 0 0 123 Khinchin Families and Hayman Class 855 Let us distinguish now between radius of convergence R being ﬁnite or not. • If R < +∞ and M < ∞, then () implies that lim f (t)< ∞ and then f t ↑ R a R < ∞. And, besides, from () we deduce also that n=0 () na R < ∞ . n=0 And, conversely, if () holds and f in K has radius of convergence R, then na R n=0 M = < ∞ . a R n=0 • If R =∞ and M < ∞, then () implies that f is a polynomial. And conversely, for a polynomial f in K of degree N , one actually has M = N . In summary, Lemma 2.2 For f (z) = a z in K with radius of convergence R > 0, we have n=0 M < ∞ if and only if R < ∞ and na R < ∞,orif R =∞ and f is a f n n=0 polynomial. ∞ ∞ n n In the ﬁrst case, we have M = ( na R )/( a R ). For a polynomial f n n n=0 n=0 f ∈ K, we have M = deg( f ). In all forthcoming applications, the exceptional cases above will not appear and we will always have M =+∞. Important notation: If M =∞, we denote by t the unique value t ∈[0, R) such f n that m(t ) = n. Observe that the t verify that lim t = R. n n→∞ n 2.1.3 Normalization and Characteristic Functions For each t ∈ (0, R), we denote the normalization of X by X : t t X − m(t ) X = for t ∈ (0, R). σ(t ) The characteristic function of X may be written in terms of the power series f as ∞ ∞ ı θ X ınθ n ınθ E(e ) = e P( X = n) = a t e t n f (t ) n=0 n=0 (2.3) ı θ f (te ) = , for t ∈ (0, R) and θ ∈ R , f (t ) while for its normalized version X we have ı θ X ıθ( X −m(t ))/σ (t ) t t E(e ) = E(e ) (2.4) ı θ X /σ (t ) −ı θ m(t )/σ (t ) = E(e ) e , for t ∈ (0, R) and θ ∈ R , 123 856 A. Cantón et al. and so, ı θ X ı θ X /σ (t ) t t E(e ) = E(e ) , for t ∈ (0, R) and θ ∈ R . 2.1.4 Scale Let f ∈ K have radius of convergence R and Khinchin family ( X ) .Fix an t t ∈[0, R) integer m ≥ 1 and let g (z) be the power series deﬁned by g (z) = f (z ). Then g m m m 1/m is also in K and g has radius of convergence R . Let (Y ) 1/m be the Khinchin family of g .Wehave t m t ∈[0, R ) 1/m Y = mX , for 0 ≤ t < R . t t ∞ n ∞ km For, if f (z) = a z , then g (z) = a z and for each k ≥ 0 and each n m k n=0 k=0 1/m 0 ≤ t < R , we have that km km a t a t k k P(Y /m = k) = P(Y = km) = = = P( X = k). t t t g (t ) f (t ) 2.1.5 Auxiliary Function F The function f does not vanish on the real interval [0, R). And so, it does not vanish in some simply connected region containing that interval. There we may consider ln f , a branch of the logarithm of f which is real on [0, R), and deﬁne the function F (z) = ln f (e ), which is holomorphic in a region containing (−∞, ln R). If f does not vanish anywhere in the disk D(0, R), then the auxiliary function F (z) is deﬁned in the whole half plane Rez < ln R. In general, the auxiliary function F (z) is deﬁned and holomorphic in the half band-like region ={s + ı θ : s < ln R, |θ | < 2/σ (e )}. This follows, for instance, see [3, Prop. 7.8], from the following lemma. ı θ Y 2 Lemma 2.3 If Y is a random variable and E(e ) = 0, then θ V(Y ) ≥ 2. ı θ Combining this lemma and formula (2.3), we deduce that f (te ) = 0, if |θ | < √ √ s+ı θ s z 2/σ (t ), and so f (e ) = 0, if |θ | < 2/σ (e ),for s < ln R. Therefore, f (e ) vanishes nowhere in . Proof of Lemma 2.3 By considering Y − E(Y ), we may assume that E(Y ) = 0. For a complex valued C function h deﬁned in R, we have that y u h( y) − h(0) − h (0) y = h (v) dvdu . 0 0 123 Khinchin Families and Hayman Class 857 ı φ y Applying this identity to h( y) = e , we see that the inequality 2 2 φ y ı φ y |e − 1 − ıyφ|≤ holds for y,φ ∈ R. Substituting Y for y, and θ for φ and taking expectations, we 2 2 2 conclude that 1 ≤ θ E(Y )/2 = θ V(Y )/2. In terms of the auxiliary function F, the mean and variance functions of the Khinchin family of f , given by (2.1), may be expressed as follows: m(t ) = F (s) and σ (t ) = F (s), fort = e and s < ln R . (2.5) 2.1.6 Some Basic Examples We now exhibit explicit formulas for the mean and variance functions and for the characteristic functions of some basic examples. (a) Let f (z) = 1 + z. In this case R =∞, and the mean and variance functions, 2 2 given by (2.1), are m(t ) = t /(1 + t ) and σ (t ) = t /(1 + t ) . For each t > 0, the variable X is a Bernoulli variable with parameter t /(1 + t ), and its characteristic function is, see formula (2.3), ı θ ı θ f (te ) 1 + te ı θ X E(e ) = = , for θ ∈ R and t > 0 , f (t ) 1 + t and thus, using formula (2.4), √ √ ıθ/ t −ı θ t te + e ı θ X E(e ) = , for θ ∈ R and t > 0 . (2.6) 1 + t (b) Let f (z) = 1/(1 − z). In this case R = 1, and the mean and variance functions 2 2 are m(t ) = t /(1 − t ) and σ (t ) = t /(1 − t ) . For each t ∈ (0, 1), the variable X is a geometric variable (number of failures until ﬁrst success) of parameter 1 − t, and its characteristic function is, see formula (2.3), ı θ f (te ) 1 − t ı θ X E(e ) = = , for θ ∈ R and t ∈ (0, 1), ı θ f (t ) 1 − te and thus, using formula (2.4), we have that 1 − t ı θ X −ı θ t E(e ) = √ e ıθ(1−t )/ t 1 − te 1 − t = √ √ , for θ ∈ R and t ∈ (0, 1). (2.7) ı θ t ıθ/ t e − te 123 858 A. Cantón et al. (c) Let f (z) = e . In this case R =∞, and the mean and variance functions are m(t ) = t and σ (t ) = t. For each t > 0, the variable X in its Khinchin family follows a Poisson distribution with parameter t, and its characteristic function is, see formula (2.3), ı θ f (te ) ı θ ı θ X t (e −1) E(e ) = = e , for θ ∈ R and t > 0 , f (t ) and thus, using formula (2.4), we have that ı θ ı θ X ı (θ / t ) E(e ) = exp t e − 1 − √ , for θ ∈ R and t > 0 · (2.8) (d) Let z n f (z) = exp(e − 1) z , for z ∈ C , n! n=0 where B (not to be confused with the nth Bernoulli number B )isthe nth Bell n n number, which counts the number of partitions of the set {1,..., n};see [9, p. 109]. t 2 Then R =∞, and the mean and variance functions are m(t ) = te and σ (t ) = t (t + 1)e . The characteristic function is given by, ı θ f (te ) ı θ ı θ X te t E(e ) = = exp e − e , for θ ∈ R and t > 0 , f (t ) and thus, using formula (2.4), we have that −t /2 ı θ e / t (t +1) t ı θ X te t t /2 E(e ) = exp e − e − ı θ e , for θ ∈ R and t > 0 . t + 1 The asymptotic formula for the B , due to Moser and Wyman [22], is dealt with in Sect. 5.2. 2.1.7 Partition Functions We next turn our attention to generating functions of partition of integers, standard and with a variety of restrictions on the admitted parts. These examples of functions in K are quite relevant in this paper. Asymptotic formulas of their coefﬁcients are dealt with in Sect. 6. The mean and variance functions or the characteristic function of the Khinchin family of these partitions functions are not as direct as in the previous examples. Closed formulas of their mean and variance functions involving series are presented in Sect. 2.2.1, and convenient approximations are exhibited in Sect. 2.2.1. 123 Khinchin Families and Hayman Class 859 (e) The ogf of partitions, the partition function, given by ∞ ∞ P(z) = = p(n) z , for z ∈ D , 1 − z j =1 n=0 is in K. The ogf Q(z) of partitions into distinct parts (which is also the ogf of partitions into odd parts), ∞ ∞ ∞ j n Q(z) = (1 + z ) = = q(n) z , for z ∈ D , 2 j +1 1 − z j =1 j =0 n=0 is also in K. Observe that P(z) Q(z) = , for z ∈ D . (2.9) P(z ) For integers a ≥ 1, b ≥ 1, the inﬁnite product P (z) = , for z ∈ D , a,b aj +b 1 − z j =0 the ogf of the partitions whose parts lie in the arithmetic progression {aj + b : j ≥ 0},isalsoin K. Observe that P ≡ P and that P ≡ Q. 1,1 2,1 (f) For integers a ≥ 1, b ≥ 0, the inﬁnite product W (z) given by W (z) = , for z ∈ D, 1 − z j =1 0 1 is also in K.Wehave W ≡ P.Also, W (z), known as the MacMahon func- 1 1 tion [20], turns out to be the ogf of plane partitions;see [9, p. 580], and Bender–Knuth [4], for a simple proof of this fact. Besides, W (z) is the ogf of partitions with parts which are ath powers of positive integers. In general, W (z) is the ogf of partitions with parts which are ath powers of positive b a integers and with number of colors j for part j ,for j ≥ 1. See Remark 2.4 for the terminology. Remark 2.4 (Colored/weighted partitions) Given a sequence (b ) of integers b ≥ j j ≥1 j 0, the inﬁnite product 1 − z j =1 123 860 A. Cantón et al. is the ogf of colored partitions with coloring sequence (b ) . A colored partition j j ≥1 of n with the above coloring sequence is an array of integers n( j , k) ≥ 0 with j ≥ 1 and 1 ≤ k ≤ b , and n = n(k, j ), i.e., of partitions whose part j may j ≥1;1≤k≤b b b come in b different colours. In the case of W , the coloring sequence is b = j ,for j j j ≥ 1. The colored partitions with the above coloring sequence are partitions in which the part j may appear in any of b available colors, with the order of the colors not mattering. These colored partitions are also called weighted partitions; the weights being the b . See Granovsky–Stark [11], and Granovsky et al. [12], where asymptotics about these (quite general) partitions are approached elegantly and systematically via Meinardus’ theorem, [21], and Khinchin families. 2.1.8 Products and Independence If f , g ∈ K, then the product h = f · g is also in K. If both f and g have radius of convergence at least R, then the product h has radius of convergence at least R.Ifthe respective Khinchin families of f , g and h are ( X ) , (Y ) and ( Z ) , t t ∈[0, R) t t ∈[0, R) t t ∈[0, R) then the law of Z is the law of the independent sum X ⊕ Y of the laws of X and Y : t t t t t P( Z = n) = P( X = k) P(Y = n − k), for each n ≥ 0 . t t t k=0 Thus, products of functions of K become, on the Khinchin family side, sums of independent variables. 2.1.9 Convergence Let ( f ) be a sequence of functions in K, all of them with radius of convergence k k≥1 at least R > 0, which converges uniformly on compacts sets of D(0, R) to a function [k] f ∈ K.Let ( X ) and ( X ) be the Khinchin families of f ,for k ≥ 1, t ∈[0, R) t t ∈[0, R) k and, respectively, of f . [k] Then, for each t ∈[0, R), we have that ( X ) converges in distribution to X k≥1 t as k →∞. [k] Also, for any moment, say, of order q ≥ 1, we have lim E(( X ) ) = E( X ), k→∞ t t since if D is the operator D = z(d/dz), then E( X ) = D ( f )(t )/ f (t ). In particular, for each t ∈[0, R), we have that lim m (t ) = m(t ) and that k→∞ k 2 2 2 2 lim σ (t ) = σ (t ), where m (t ), σ (t ) and m(t ), σ (t ) are, respectively, the k→∞ k k k mean and variance functions of f and f . 123 Khinchin Families and Hayman Class 861 2.2 Infinite Products Let ( f ) be a sequence of functions in K all with radius of convergence at least j j ≥1 R > 0. Assume that the inﬁnite product f (z) = f (z) j =1 converges absolutely and uniformly on compact subsets of D(0, R). The product f has non-negative Taylor coefﬁcients. Since f (0) = 0, for each j, we have f (0) = 0, and actually, f (0)> 0. Moreover, from f (t ) f (t ) = , for t ∈[0, R), f (t ) f (t ) j =1 we see that f the inﬁnite product is non-constant. Thus the product function f is in K and its power series has radius of convergence at least R. Let ( X ) be the Khinchin family of f and let m(t ) = E( X ) and σ (t ) = t t ∈(0, R) t V( X ) be its mean and variance functions. For each j ≥ 1, the Khinchin family of f (z) is denoted by X and its mean and variance functions by m (t ) and σ (t ),for j j ,t j t ∈[0, R). Since f (z) converges uniformly as N →∞ on compact subsets of D(0, R), j =1 we have that, for each t ∈ (0, R),the sum X converges in law to X ,as j ,t t j =1 N →∞, and that for the means and variances we have ∞ ∞ 2 2 m(t ) = m (t ) and σ (t ) = σ (t ), for each t ∈[0, R). j =1 j =1 2.2.1 Partition Functions as Inﬁnite Products As an illustration of the above, we consider now partitions and partitions into distinct parts. Consider ﬁrst the partition function P(z) = , for |z| < 1. 1 − z j =1 Let ( X ) be its Khinchin family. And let (Y ) be the Khinchin family of t t ∈[0,1) t t ∈[0,1) 1/(1 − z). Because of scaling properties, see Sect. 2.1.4, we have that X = jY , for t ∈ (0, 1), j =1 123 862 A. Cantón et al. and each Y is a geometric variable with parameter 1 − t . For its mean and variance functions, m(t ) and σ (t ), we have that ∞ ∞ j 2 j jt j t m(t ) = and σ (t ) = , for t ∈ (0, 1). j j 2 1 − t (1 − t ) j =1 j =1 We will obtain closed form approximations of m(t ) and σ (t ) as t ↑ 1 in Sect. 6.1. If ( X ) is the Khinchin family of the ogf t t ∈[0,∞) Q(z) = (1 + z ) j =1 of partitions into distinct parts, we then have, because of scaling properties, see Sect. 2.1.4, that X = jY j , for t > 0 , j =1 j j and each Y j is a Bernoulli variable with parameter t /(1 + t ). For its mean and variance functions we have that ∞ ∞ j 2 j jt j t m(t ) = and σ (t ) = , for t ∈ (0, 1). j j 2 1 + t (1 + t ) j =1 j =1 Analogous expressions for the mean and variance functions of other partition func- tions like P or W will appear later in Sect. 6.1. a,b a,b 2.3 Hayman’s Identity For f (z) = a z ∈ K, Cauchy’s formula for the coefﬁcient a in terms of the n n n=0 characteristic function of its Khinchin family ( X ) reads t t ∈[0, R) f (t ) ı θ X −ı θ n a = E(e ) e dθ, for each t ∈ (0, R) and n ≥ 1 . 2π t |θ |<π In terms of the characteristic function of the normalized variable X , it becomes, for each t ∈ (0, R) and n ≥ 1, f (t ) ı θ X −ıθ(n−m(t ))/σ (t ) a = E(e ) e dθ. 2π t σ(t ) |θ |<π σ (t ) 123 Khinchin Families and Hayman Class 863 If M =∞, we may take for each n ≥ 1 the (unique) radius t ∈ (0, R) so that f n m(t ) = n, to write f (t ) n ˘ ı θ X a = E(e ) dθ, for each n ≥ 1 , (2.10) 2π t σ(t ) |θ |<π σ (t ) n n which we call Hayman’s identity. If we write this identity (2.10)as a 2π t σ(t ) n n ˘ n ı θ X = E(e ) dθ f (t ) n |θ |<π σ (t ) we see that an asymptotic formula for a follows if one is able to determine the ı θ X behaviour of E(e ) as t → R. (Recall that t → R,as n →∞, see Sect. 2.1.2.) Of course, once that has been achieved one still needs to determine the t and also appropriate expressions for the f (t ) and the σ(t ), which, in general, is not so direct. n n In any case, the plan above is actually the route to go. 3 Gaussian Khinchin Families In this section we discuss the notions of Gaussian and strongly Gaussian Khinchin families, obtain Hayman’s asymptotic formula for its coefﬁcients and, ﬁnally, intro- duce the class of Hayman functions, which amounts to a criterion for being strongly Gaussian. As we have seen, Theorem A, strongly Gaussian power series are Gaussian; and as we shall see, Theorem 3.8, power series in the Hayman class are strongly Gaussian. 3.1 Gaussian Khinchin Families Deﬁnition 3.1 (Gaussian power series) A power series f ∈ K and its Khinchin family ( X ) are termed Gaussian if X converges, as t ↑ R, in distribution to the t t ∈[0, R) t standard normal or, equivalently, if ı θ X −θ /2 lim E(e ) = e , for each θ ∈ R . t ↑ R For the exponential function f (z) = e , we deduce directly by taking the limit as t ↑ ∞ in formula (2.8) of the characteristic function of its normalized family, with θ ∈ R ﬁxed, that e is Gaussian. This fact means in particular that the normalized version of the Poisson random variable X , that is, ( X − t )/ t, converges in distribution to the t t standard normal. 123 864 A. Cantón et al. The function f (z) = 1/(1 − z) is not Gaussian. By taking limits in formula (2.7) as t ↑ 1, with θ ∈ R ﬁxed, we obtain that −ı θ ı θ X lim E(e ) = , for each θ ∈ R . t ↑1 1 − ı θ This actually means that X converges in distribution towards a variable Z, where Z + 1 is an exponential variable of parameter 1. The function f (z) = 1 + z is not Gaussian, either. By taking limits in formula (2.6) as t ↑∞, with θ ∈ R ﬁxed, we obtain that ı θ X lim E(e ) = 1 , for each θ ∈ R , t ↑∞ In other terms, X converges in distribution towards the constant 0. 3.1.1 A Criterion for Gaussianity The following simple criterion for gaussianity of functions f ∈ K non-vanishing in D(0, R), and in terms of the auxiliary function F of Sect. 2.1.5, is implicit in Hayman’s paper, see [16,Lem.4]. Theorem 3.2 (A criterion for gaussianity) If f ∈ K has radius of convergence R > 0 and vanishes nowhere in D(0, R), and if for the auxiliary function F one has sup | F (s + i φ)| φ∈R lim = 0 , (3.1) 3/2 s↑ln R F (s) then f is Gaussian. Proof. Since f vanishes nowhere in D(0, R), the auxiliary function F (z) = ln f (e ) is deﬁned and is holomorphic in the whole half-plane {z ∈ C : Rez < ln R}.Wehave, for s < ln R and θ ∈ R, that 2 3 θ |θ | F (s + ıθ) − F (s) − F (s)ı θ + F (s) ≤ sup F (s + i φ) · 2 6 φ∈R 2 s s Since F (s) = m(e ) and F (s) = σ (e ) for each s < ln R, writing e = t and substituting θ by θ/σ (t ) we deduce that sup F (s + i φ) m(t ) θ |θ | φ∈R ıθ/σ (t ) ln f (te ) − ln f (t ) − ı θ + ≤ , σ(t ) 2 σ (t ) 6 or, equivalently, for t = e with s < ln R and θ ∈ R, 123 Khinchin Families and Hayman Class 865 sup F (s + i φ) θ |θ | ˘ φ∈R ı θ X ln E(e ) + ≤ · (3.2) 3/2 2 F (s) 6 3.1.2 Some Applications of the Use of the Criterion of Theorem 3.2 Recall the auxiliary function F (z) = ln f (e ) from Sect. 2.1.5. For the exponential function f (z) = e , the auxiliary function is given by F (z) = e , and the gaussianity of f follows readily from Theorem 3.2. j R(z) Similarly, if R(z) = b z is a polynomial of degree N such that e ∈ K, j =0 R(z) z then e is Gaussian. For in this case F (z) = R(e ), and for z = s + ı φ we have that N N 3 jz 3 js Ns | F (z)|= b j e ≤ |b | j e = O(e ), j j j =0 j =0 while 2 js 2 Ns () F (s) = b j e ∼ b N e as s ↑∞ , j N j =0 and gaussianity follows from Theorem 3.2. Observe that () implies that b is real and b > 0; besides, since e ∈ K, the coefﬁcients of the polynomial R must be real numbers. For the partition function P(z) = 1/(1 − z ), the auxiliary function F may j =1 be written as kj z F (z) = e , for Rez < 0 . j ,k≥1 For z such that Rez < 0 and integer q ≥ 1, the qth derivative of F is given by (q) q−1 q kj z F (z) = k j e . j ,k≥1 For z = s + ı φ, with s < 0 and φ ∈ R, we then have that 2 3 kj s F (s + ı φ) ≤ k j e = F (s). j ,k≥1 Condition (3.1) requires one to check that F (−s) lim = 0 . 3/2 s↓0 F (−s) 123 866 A. Cantón et al. Let us see. For s > 0 and integer q ≥ 1, we have that q+1 (q) q−1 q −k( js) s F (−s) = k s ( js) e , k≥1 j ≥1 and so ∞ ∞ q+1 (q) q−1 q −kx q − y lim s F (−s) = k x e dx = y e dy s↓0 k 0 0 k≥1 k≥1 = ζ(2)(q + 1), and, thus, (q) F (−s) ∼ ζ(2)(q + 1) , as s ↓ 0 . (3.3) q+1 In particular, F (−s) 3 1/2 ∼ s , as s ↓ 0 . 3/2 F (−s) 2ζ(2) We conclude that the partition function P is Gaussian. Likewise, one readily veriﬁes that the ogfs Q and P are Gaussian. a,b For the inﬁnite product W (z) with a ≥ 1, b ≥ 0, let F (z) be the corresponding a,b auxiliary function. The following asymptotic formula for the qth derivative of F : a,b 1 b + 1 b + 1 1 (q) F (−s) ∼ ζ 1 + q + , as s ↓ 0 , (3.4) a,b q+(b+1)/a a a a s may be obtained with an argument very much like the one above for the partition function P. We deduce that F (−s) a,b (b+1)/(2a) s , as s ↓ 0 , 3/2 F (−s) a,b thus showing that each W , with a ≥ 1, b ≥ 0, is Gaussian. 3.1.3 A Criterion of Gaussianity for f = e in Terms of g Not unusually, we start with a power series g with non-negative coefﬁcients and are actually interested in its exponential f = e . The symbolic method of Combinatorics, see Flajolet–Sedgewick [9, Ch. II], gives that if g is the exponential generating function (egf) of a combinatorial class G of labelled objects with g(0) = 0, then e is the egf of the class of sets formed with the objects of G. For instance, e −1 • e is the egf of the class of sets of non-empty sets (or partitions of a set), see [9, p. 107], to be discussed in Sect. 5.2.1, 123 Khinchin Families and Hayman Class 867 ze • e is the egf of the class of sets of pointed sets (or idempotent functions), see [9, p. 131], to be discussed in Sect. 5.2.2, z/(1−z) • e is the egf of the class of sets of permutations (or ‘fragmented permuta- tions’), see [9, p. 125], to be discussed in Sect. 5.3. Observe that in the ﬁrst two instances R =∞, while in the last one, R = 1. The following theorem, a consequence of Theorem 3.2, exhibits conditions on g which imply that f = e is Gaussian. Theorem 3.3 (A criterion of gaussianity for f = e in terms of g) Let f ∈ K be such that f = e , where g has radius of convergence R > 0 and non-negative coefﬁcients. If g is a polynomial of degree 1 or g satisﬁes g (t ) lim = 0 , (3.5) 3/2 t ↑ R g (t ) then f is Gaussian. The three examples above are readily seen to be Gaussian as a consequence of Theorem 3.3. Proof We will assume that g is not a polynomial of degree 1 and that (3.5) is satisﬁed. z z The auxiliary function F is given by F (z) = ln f (e ) = g(e ),for Rez < ln R. Write g(z) as g(z) = b z , with b ≥ 0, for |z| < R. n n n=0 ( j ) j nz For an integer j ≥ 1, we have that F (z) = n b e ,for Rez < ln R.If n=1 s 3 9 we write t = e , with s < ln R, and let φ ∈ R, by using that n ≤ n(n − 1)(n − 2) for n ≥ 3, we have that ∞ ∞ 3 n 2 3 n−3 | F (s + ı φ)|≤ n b t ≤ b t + 8b t + t n(n − 1)(n − 2)b t n 1 2 n n=1 n=3 2 3 ≤ b t + 8b t + t g (t), 1 2 and by observing that n ≥ n(n − 1), also that F (s) ≥ t g (t). Therefore, 2 9 3 sup | F (s + ı φ)| b t + 8b t + t g (t ) 1 2 φ∈R () ≤ . 3/2 3 3/2 F (s) t g (t ) If R =∞, condition (3.1) of Theorem 3.2 follows from (), hypothesis (3.5) and the fact that lim g (t)> 0. t ↑ R If R < +∞, then () lim g (t ) =+∞. For otherwise, because of (3.5), we t ↑ R would have that lim g (t ) = 0, so that g ≡ 0 and g would be a polynomial of t ↑ R degree at most 2 and R =∞.From (), () and condition (3.5), we conclude that condition (3.1) of Theorem 3.2 is satisﬁed. 123 868 A. Cantón et al. 3.2 Strongly Gaussian Khinchin Families Following Báez-Duarte [2], we introduce the notion of strongly Gaussian power series. Deﬁnition 3.4 (Strongly Gaussian power series) A power series f ∈ K with radius of convergence R and its Khinchin family ( X ) are termed strongly Gaussian t t ∈[0, R) if ı θ X −θ /2 (a) lim σ(t ) =+∞ and (b) lim E(e ) − e dθ = 0 . t ↑ R t ↑ R |θ |<π σ (t ) The exponential function f (z) = e is strongly Gaussian. Recall that in this case σ(t ) = t. Strong gaussianity of e follows from its gaussianity and dominated convergence using the bound 2 2 ı θ X t (cos(θ / t )−1) −2θ /π E(e ) = e ≤ e , for θ ∈ R, t > 0 such that |θ | <π t . Theorem A below is both a local and a non-local central limit theorem satisﬁed by strongly Gaussian power series. It appears in Hayman’s [16] as Theorem I and Theorem II under the stronger hypothesis that f is in the Hayman class (to be discussed shortly in Sect. 3.3), but the proofs in [16] work for strongly Gaussian power series. Theorem A (Hayman’s central limit theorem) If f (z) = a z in K is strongly n=0 Gaussian, then a t 2 2 −(n−m(t )) /(2σ (t )) lim sup 2πσ (t ) − e = 0 . (3.6) t ↑ R f (t ) n∈Z Besides, lim P( X ≤ b) = (b), for every b ∈ R . t ↑ R And so, X converges in distribution towards the standard normal, and f is Gaussian. In this statement a = 0, for n < 0. By considering n =−1in (3.6), it follows that lim (m(t )/σ (t )) =+∞. In particular, this means that M given by (2.2)is t ↑ R f M =∞ for every f ∈ K that is strongly Gaussian. Theorem B (Hayman’s asymptotic formula) If f (z) = a z in K is strongly n=0 Gaussian, then 1 f (t ) a ∼ √ , as n →∞ , (3.7) t σ(t ) 2π n where t is given by m(t ) = n, for each n ≥ 1. n n 123 Khinchin Families and Hayman Class 869 This asymptotic formula is obtained by using Hayman’s identity (2.10) as follows. Write 2π a t σ(t ) 1 n n ˘ ı θ X − 1 = √ E(e ) dθ − 1 f (t ) n 2π |θ |<π σ (t ) 1 1 2 2 ı θ X −θ /2 −θ /2 = √ E(e ) − e dθ − √ e dθ, 2π 2π |θ |<π σ (t ) |θ |≥πσ (t ) n n and observe that conditions b) and a), respectively, of strong gaussianity (Deﬁni- tion 3.4), show that the ﬁrst and second terms of the expression above tend to 0 as n →∞. Actually, if ω is a good approximation of t , in the sense that n n m(ω ) − n lim = 0 , n→∞ σ(ω ) then 1 f (ω ) a ∼ √ , as n →∞ . (3.8) ω σ(ω ) 2π n Formula (3.8) (and also formula (3.7)) follows immediately from (3.6) of Theo- rem A. For the exponential f (z) = e one has m(t ) = t and σ(t ) = t,for t ≥ 0, and t = n for n ≥ 1. The asymptotic formula above gives 1 1 e ∼ √ √ , as n →∞ , n! n n 2π that is Stirling’s formula. Remark 3.5 The function f (z) = e is Gaussian, because of Theorem 3.3. Its variance 2 2 function σ (t ) = 4t tends towards ∞ as t ↑∞,but f is not strongly Gaussian, since its Taylor coefﬁcients of odd order are null and do not satisfy the asymptotic formula (3.7). 3.2.1 Báez-Duarte Substitution In general, precise expressions for the t are rare, since inverting m(t ) is usually compli- cated. But, fortunately, in practice, one can do with a certain asymptotic approximation due to Báez-Duarte [2]. This is the content of Theorem C below. Suppose that f ∈ K is strongly Gaussian. Assume that m (t ) is continuous and monotonically increasing to +∞ in [0, R), and that m (t ) is a good approximation of m(t ) in the sense that m(t ) − m (t ) lim = 0 . (3.9) t ↑ R σ(t ) Let τ be deﬁned by m (τ ) = n, for each n ≥ 1. n n 123 870 A. Cantón et al. Theorem C (Substitution) With the notations above, if f (z) = a z in K is n=0 strongly Gaussian and (3.9) is satisﬁed, then 1 f (τ ) a ∼ √ , as n →∞ . τ σ(τ ) 2π n Moreover, if σ(t ) is such that σ(t ) ∼ σ(t ) as t ↑ R, we may further write 1 f (τ ) a ∼ , as n →∞ . (3.10) τ σ(τ ) 2π n Theorem C follows readily from (3.8). In Sect. 6.1, we will exhibit appropriate approximations m and σ of the mean and 2 b variance functions m and σ of the ogfs of partitions P, Q, P and W , satisfying a,b in each case condition (3.9). 3.3 Hayman Class Strongly Gaussian functions in K have excellent asymptotic properties: the central limit Theorem A and the asymptotic formula (3.7) for its coefﬁcients. Thus far, f (z) = e is the only strongly Gaussian function encountered here. The class of Hayman consists of power series f in K which satisfy some concrete and veriﬁable conditions which imply that f is strongly Gaussian, see Theorem 3.8. Deﬁnition 3.6 (Hayman class) A power series f ∈ K is in the Hayman class if (variance condition) : lim σ(t ) =∞ , (3.11) t ↑ R and if for a certain function h:[0, R) → (0,π ], which we refer to as the cut (between a major arc and a minor arc), there hold ı θ X θ /2 (major arc) : lim sup E(e ) e − 1 = 0, (3.12) t ↑ R |θ |≤h(t)σ(t ) ı θ X (minor arc) : lim σ(t ) sup |E(e )|= 0 , (3.13) t ↑ R h(t )σ (t )<|θ |≤πσ (t ) The cut h of a f in the Hayman class is not uniquely determined. The functions in the Hayman class are called admissible by Hayman [16]; they are also called Hayman-admissible, or just H-admissible. Unlike other accounts (including Hayman’s), we do not allow a ﬁnite number of the coefﬁcients of the power series to be negative. For f in the Hayman class, the characteristic function of X is uniformly approxi- −θ /2 mated by e in the major arc, while it is uniformly o(1/σ (t )) in the minor arc. 123 Khinchin Families and Hayman Class 871 Observe that condition (3.13) may be written in terms of f or X as the requirement that ı θ | f (te )| ı θ X lim σ(t ) sup = 0 or lim σ(t ) sup E(e ) = 0 . t ↑ R f (t ) t ↑ R h(t )<|θ |≤π h(t )<|θ |≤π Lemma 3.7 If f ∈ K is in the Hayman class with cut h, then lim h(t)σ (t ) =∞ . t ↑ R ı θ X θ /2 t t Proof For θ = h(t)σ (t ), condition (3.12) implies that lim E(e ) e = 1; t t ↑ R ı θ X θ /2 t t while condition (3.13) implies that lim σ(t ) E(e ) = 0 . Thus, lim e /σ (t ) t ↑ R t ↑ R =∞ . As lim σ(t ) =∞, the conclusion follows. t ↑ R The conditions for a power series to be in the Hayman class amount to a criterion to be strongly Gaussian: Theorem 3.8 Power series in the Hayman class are strongly Gaussian. Proof Let us denote ı θ X −θ /2 I = E(e ) − e dθ, for t ∈ (0, R). |θ |<π σ (t ) Divide the integral I into an integral J over the major arc and an integral K over the t t t minor arc: ı θ X −θ /2 J = E(e ) − e dθ, |θ |≤h(t )σ (t ) ı θ X −θ /2 K = E(e ) − e dθ. h(t )σ (t )<|θ |≤πσ (t ) Bound K as ı θ X −θ /2 K ≤ 2πσ (t ) sup E(e ) + e dθ. h(t )σ (t )<|θ |≤πσ (t ) |θ |≥h(t )σ (t ) The ﬁrst summand of the bound of K tends to 0, as t ↑ R, because of the hypoth- esis (3.13), while the second summand tends to 0, as t ↑ R, since, as Lemma 3.7 dictates, lim h(t )σ (t ) =+∞. t ↑ R Bound J as 2 2 −θ /2 ı θ X θ /2 J = e E(e ) e − 1 dθ |θ |≤h(t )σ (t ) 2 2 −θ /2 ı θ X θ /2 ≤ e dθ sup E(e ) e − 1 R |θ |≤h(t )σ (t ) 123 872 A. Cantón et al. This bound of J tends to 0 as t ↑ R, by virtue of the hypothesis (3.12). As a consequence of Theorem 3.8, for asymptotic estimation of coefﬁcients of func- tions of the Hayman class, one may use the more ﬂexible asymptotic formula (3.10) of Báez-Duarte instead of the proper Hayman’s asymptotic formula (3.7). This fact will be crucial later, particularly for the arguments pertaining the partition functions of Sect. 6. 3.3.1 Combination and Perturbation of Functions of Hayman Class Maybe the reason for the relevance of the Hayman class in combinatorial questions is that it enjoys some closure and small perturbation properties. For instance, see, of course, Hayman [16]: (a) If f and g are in the Hayman class, then the product f · g is in the Hayman class, see [16, Thm. VII]. (b) If f is in the Hayman class, then e is in the Hayman class, see [16, Thm. VI]. (c) If f is in the Hayman class and if R is a polynomial in K, then the product R · f is in the Hayman class, see [16, Thm. VIII]. N n (d) Let B(z) = b z be a non-constant polynomial with real coefﬁcients such n=0 that for each d > 1 there exists m, not a multiple of d, such that b = 0, and such B B that if m(d) is the largest such m, then b > 0. If e is in K, then e is in the m(d) Hayman class, see [16,Thm.X]. 4 Exponentials and Hayman Class Let g be a non-constant power series with non-negative coefﬁcients and radius of convergence R > 0. The exponential f = e of such a g is in K. As mentioned above, see Sect. 3.1.3, this setting is quite usual in combinatorial questions. Write g(z) = b z , with b ≥ 0for n ≥ 0, and let f ∈ K be given by n n n=0 f = e . We will exhibit in Theorem 4.1 below conditions on the function g which ensure that f is in the Hayman class so that we can obtain an asymptotic formula for its coefﬁcients. Equipped with Theorem 4.1 below and the substitution Theorem C, we will discuss in Sects. 5.1, 5.2 and 5.3 a number of asymptotic formulae for coefﬁcients, mostly (but not only) of combinatorial interest, for functions f of the form f = e with non-constant g with non-negative coefﬁcients. Let ( X ) be the Khinchin family of f . The mean and variance functions of f , t t ∈[0, R) written in terms of g,are m(t ) = tg (t ) and σ (t ) = tg (t ) + t g (t), for t ∈ (0, R). Since g has non-negative coefﬁcients, the variance function σ (t ) of f is increasing in [0, R). The auxiliary function F of f is the function F (z) = g(e ),for z such that Rez < ln R. 123 Khinchin Families and Hayman Class 873 The variance condition (3.11) for belonging to the Hayman class in terms of g becomes lim tg (t ) + t g (t ) =+∞ . t ↑ R We want a cut function h for f . Let us start with the major arc. The bound (3.2) gives that sup F (s + ı φ) θ |θ | ˘ φ∈R ı θ X s ln E(e ) + ≤ , for t = e and s < ln R . 2 σ (t ) 6 The proof of Theorem 3.3 gives also that 2 3 | F (s + ı φ)|≤ b t + 8b t + t g (t), for t = e , s < ln R and φ ∈ R . 1 2 Deﬁne 2 3 ω (t ) = b t + 8b t + t g (t), for t ∈ (0, R). (4.1) g 1 2 z |z| Consider a potential cut function h:[0, R) → (0,π ]. Using that |e − 1|≤|z|e for all z ∈ C, we obtain that ı θ X θ /2 (t ) sup E(e ) e − 1 ≤ (t ) e , |θ |≤h(t)σ(t ) where (t ) is given by (t ) = ω (t ) h(t ) , for t ∈ (0, R). So, for h to fulﬁll condition (3.12) on the major arc, it is enough to have that lim ω (t ) h(t ) = 0 . t ↑ R In terms of h and g, condition (3.13) on the minor arc becomes ı θ lim σ(t ) exp sup Reg(te ) − g(t ) = 0 . t ↑ R h(t )≤|θ |≤π Thus, Theorem 4.1 Let g be a non-constant power series with radius of convergence R and non-negative coefﬁcients satisfying the variance condition (variance condition) : lim tg (t ) + t g (t ) =+∞ . (4.2) t ↑ R 123 874 A. Cantón et al. If there is a cut function h:[0, R) → (0,π ] satisfying (major arc) : lim ω (t ) h(t ) = 0 , (4.3) t ↑ R and ı θ (minor arc) : lim σ(t ) exp sup Reg(te ) − g(t ) = 0 , (4.4) t ↑ R h(t )<|θ |≤π then f = e is in the Hayman class. The function ω appearing in the major arc condition is given by the formula (4.1). Regarding the variance condition. For ﬁnite R, the variance condition (4.2) becomes lim g (t ) =∞. t ↑ R For R =∞, the variance condition (4.2) is always satisﬁed, since otherwise we would have that lim g (t ) = 0 and g would be constant. t →∞ Regarding the major arc condition. For ﬁnite R, the major arc condition (4.3) becomes lim g (t ) h(t ) = 0. t ↑ R For R =∞,if g(z) is not a polynomial, the condition on the major arc (4.3) reduces to lim t g (t ) h(t ) = 0 , t →∞ while if g is a polynomial of degree k ≥ 2, then (4.3) reduces to k 3 lim t h(t ) = 0 . t →∞ 4.1 Some Lemmas for Minor Arc Estimates In applications of Theorem 4.1, the variance and major arc condition for a cut for f = e are usually straightforward to verify. It is always the minor arc which requires the most work. We collect next a few elementary estimates for the functions 1/(1 − z) on |z|= t for t ∈ (0, 1) and integer k ≥ 1 which will prove useful in a number of veriﬁcations of the minor arc condition (4.4) of Theorem 4.1. Lemma 4.2 For t ∈ (0, 1) and θ ∈ R, −1 2 2 θ 4t sin 1 − t (1 − t ) = = 1 + . ı θ 2 2 1 − te (1 − t ) − 2t (cos θ − 1) (1 − t ) 123 Khinchin Families and Hayman Class 875 Lemma 4.3 Let ω ∈ (0,π) and t ∈ (0, 1). Then 1 − t 1 − t 1 − t 1 − t sup Re = Re and sup = . ı θ ı ω ı θ ı ω 1 − te 1 − te 1 − te 1 − te ω≤|θ |≤π ω≤|θ |≤π Proof The Möbius transformation T (z) = 1/(1 − z) carries the circle |z|= r, with r ∈ (0, 1), onto the circle orthogonal to the positive real axis through 1/(1 + r ) and ı φ 1/(1 − r ). For each r ∈ (0, 1), we have that Re((1 − r )/(1 − re )) decreases from 1to (1 − r )/(1 + r ) as φ increases from 0 to π (and as φ decreases from 0 to −π). Monotonicity of modulus follows from Lemma 4.2. Lemma 4.4 Let h(t ) be deﬁned in the interval (0, 1) with values in (0,π) and such that lim h(t )/(1 − t ) = 0. Let k be an integer k ≥ 1. Then we have t ↑1 (1 − t ) 1 − t k lim − 1 =− · ıh(t ) t ↑1 h(t ) 1 − te 2 Proof. Lemma 4.2 and lim h(t ) = 0 give that t ↑1 1 − t () lim = 1 . ıh(t ) t ↑1 1 − te Lemma 4.2 gives that h(t ) 4t sin 1 − t 2 () 1 − ∼ , as t → 1 . ıh(t ) 2 1 − te (1 − t ) The case k = 2 follows from (). The general case follows from combining (),the case k = 2 and that y − 1 k lim = · y→1 y − 1 2 5 Some Examples of Exponentials in the Hayman Class In this section we verify that relevant examples of f = e are in the Hayman class. Some of them have been mentioned already, see the discussion of Sect. 3.1.3, like z z g(z) = e − 1 (for non-empty sets) or g(z) = ze (for pointed sets). We shall consider too the example of a polynomial g with non-negative coefﬁcients or the case g(z) = (1 + z)/(1 − z). The function g in all these cases has a closed formula which facilitates the veriﬁ- cation of the conditions of Theorem 4.1, and thus the corroboration that f is in the 123 876 A. Cantón et al. Hayman class and that Hayman’s asymptotic formula (3.7) applies. Besides, the mean function m(t ) and the variance function σ(t ) have as well, in all these cases, closed formulas, and the asymptotic behaviour of t and f (t ) and σ(t ) appearing is (3.7) n n is quite direct. 5.1 Exponential of a Polynomial Let B(z) = b z be a polynomial of degree N ≥ 1 with non-negative coef- n=0 ﬁcients. Assume further that gcd{1 ≤ n ≤ N : b > 0}= 1. We will check that f = e is in the Hayman class. For this aim, we could appeal to the general result d) of Hayman of Sect. 3.3.1. But the present case of non-negative coefﬁcients may be neatly cast as a consequence of Theorem 4.1, and we include a proof. N j Proposition 5.1 Let B(z) = b z be a polynomial of degree N ≥ 1 with non- j =0 negative coefﬁcients and such that q gcd{1 ≤ n ≤ N : b > 0}= 1 . (5.1) B(z) Then, f (z) = e is in the Hayman class. Condition (5.1) is indispensable. In fact, if q > 1, then f = e is not strongly Gaussian, since then f (z) = H (z ), for a certain entire power series H, and the coef- ﬁcients of f can not satisfy Hayman’s asymptotic formula (3.7) as strongly Gaussian functions do. Proof As mentioned above, since the radius of convergence of B(z) is R =∞,the variance condition (4.2) is satisﬁed. −α We verify next that h(t ) given by h(t ) = min{π/ N , t }, with α ∈ ( N /3, N /2), satisﬁes conditions (4.3) and (4.4) of Theorem 4.1 for a cut. Fix α in that interval. Condition (4.3) on the major arc reduces, as we have mentioned above, to N 3 lim t h(t ) = 0 , t →∞ which is satisﬁed since α> N /3. Now, the minor arc. Notice that ı θ n Re B(te ) − B(t ) = b t (cos(nθ) − 1). n=1 All the summands above are non-positive. We distinguish now between θ’s close to and away from the N th roots of unity. ı θ Denote T ={e : θ ∈ R}= ∂ D. For t > 0 and 1 ≤ j < N , deﬁne the arcs in T 2π j ı θ I (t ) = e ∈ T : θ − < h(t ) . 123 Khinchin Families and Hayman Class 877 and for t > 0 deﬁne also ı θ I (t ) = e ∈ T :|θ | < h(t ) . These I (t ),0 ≤ j < N are arcs in T of length 2h(t ) around the N th roots of unity. I is the major arc. Since h(t ) ≤ π/ N , the arcs I (t ) are disjoint. 0 j ı θ ˆ ˆ Denote also by I (t ) the arc I (t ) = e ∈ T :|θ | < Nh(t ) . 0 0 ı θ Let N ={1 ≤ n ≤ N : b > 0}. We will use that, since gcd(N ) = 1, no e with ı θ θ ∈ R, except e = 1, is simultaneously a nth root of unity for all n ∈ N . ı2π j / N For each 1 ≤ j < N , there is n ∈ N so that e , the center of the arc I (t ),is j j not a n th root of unity. Since lim h(t ) = 0, there is t > 1 and, for each 1 ≤ j < N , j t ↑∞ 0 a δ > 0, so that ı θ b (cos(n θ) − 1) ≤−δ , for e ∈ I (t ) and t ≥ t . n j j j 0 Let = min{δ , 1 ≤ j < N } and M = min N . Then n M ı θ b t (cos(n θ) − 1) ≤−t , for e ∈ I (t ) and 1 ≤ j < N and t ≥ t . n j j 0 N −1 ı θ ıN θ For e ∈ T \ I (t ) ∪ I (t ) we have that e ∈ T \ I (t ) and, thus, using j 0 0 j =1 2 2 the inequality 1 − cos x ≥ 2x /π , valid for x ∈[−π, π ], that 2 2 cos( N θ) − 1 ≤− N h(t ) , and so that N N 2 b t (cos( N θ) − 1) ≤− t h(t ) , 2 2 with = (2/π ) N b . Therefore, if h(t ) ≤|θ|≤ π and t > t , we have that ı θ n M N 2 Re B(te ) − B(t ) = b t (cos(nθ) − 1) ≤ max −t , −t h(t ) . n=1 Since σ(t ) grows just as a power of t, condition (4.4) on the minor arc is satisﬁed since α< N /2. We conclude that e is in the Hayman class. 5.1.1 Frequency of Permutations of Given Order As a ﬁrst example of the efﬁcient alliance of strong gaussianity with the substitution Theorem C, we now give, for any ﬁxed integer k ≥ 1 and as n →∞, an asymptotic formula for the number A (n) of permutations σ ∈ S such that σ is the identity. k n Observe, for instance, that A (n) is the number of involutions in S . The particular 2 n 123 878 A. Cantón et al. cases k = 2 and k prime of this asymptotic formula are due to Moser–Wyman, [23,24]. For general k, the formula is essentially due to Wilf [28]. The permutations counted by A (n) are those σ ∈ S so that all the cycles in its k n cycle decomposition have lengths which divide k. Thus, the symbolic method, see [9, p. 124], gives us that the egf I (z) of these permutations is ⎛ ⎞ A (n) 1 n d ⎝ ⎠ I (z) = z = exp z . n! d n=0 1≤d|k Denote R (z) = z . 1≤d|k As I (z) is the exponential of the polynomial R (z) and gcd{d ≥ 1 : d | k}= 1, we k k have by Proposition 5.1 that I (z) is in the Hayman class and, in particular, that I (z) k k is strongly Gaussian. By applying the asymptotic formula (3.10) of the substitution Theorem C, we will obtain the asymptotic formulae (5.2) and (5.3)for A (n), distinguishing between k odd or k even. Denote by m (t ) and σ (t ),for t > 0, the mean and variance functions of the Khinchin family of I (t ). Observe that d 2 d m (t ) = t and σ (t ) = dt , for t > 0 . 1≤d|k 1≤d|k k/2 We take σ (t ) = kt ,for t > 0. Observe that σ (t ) ∼ σ (t ),as t →∞. k k k To properly deﬁne an approximation m (t ) of m (t ), we distinguish between k even k k and k odd. (a) Case k odd. The divisors d of k, different from k, satisfy d ≤ k/3. Thus m (t ) = k k/3 k t + O(t ),as t →∞, and if we deﬁne m (t ) = t , then k/3 m (t ) − m (t ) = O(t ) = o( σ (t )) , as t ↑∞ . k k k 1/k For n ≥ 1, we take τ = n , so that m (τ ) = n. Strong gaussianity and the n k n substitution Theorem C give for k ≥ 1, odd and ﬁxed, that 1/k R (n ) A (n) 1 1 e ∼ √ √ √ , as n →∞ . (5.2) n/k n! n n 2π k (b) Case k even. Now a divisor of k is k/2, which is the order of σ (t ).Wehave k k/2 k/3 m (t ) = t + t + O(t ),as t ↑∞, and we take 1 1 k k/2 k/2 m (t ) = t + t + = t + , for t > 0 . 4 2 123 Khinchin Families and Hayman Class 879 Thus, k/3 m (t ) − m (t ) = O(t ) = o( σ (t )) , as t ↑∞ . k k k 2/k Deﬁne τ = ( n − 1/2) , so that m (τ ) = n, for each n ≥ 1. Since n k n √ √ 1 1 n ln n − = n ln n + n ln 1 − √ 2 2 n n 1 1 1 = ln n − n √ + + O 3/2 2 2 n 8n n n n 1 = ln n − − + o(1), as n →∞ , 2 2 8 and so n/2 − n/2 −1/8 n − ∼ n e e , as n →∞ , we have that n n/k − n/k −1/(4k) τ ∼ n e e , as n →∞ . Besides, √ √ k/2 σ (τ ) = k τ ∼ k n , as n →∞ . k n n Strong gaussianity and substitution give for k ≥ 1, even and ﬁxed, that 1/k 1/(4k) R ((n− n+1/4) ) A (n) 1 e e n/k ∼ √ √ e √ , as n →∞ . (5.3) n/k n! n n 2π k Using (5.2) and (5.3), ones sees that if q > k, then A (n) 1 1 ln = − n ln n + O(n), as n →∞ , A (n) q k and, thus, irrespective of the parity of q and k, A (n) () lim = 0 , n→∞ A (n) Besides, lim A (n)/n!= 0 and lim A (n) =∞. n→∞ k n→∞ k For the number B (n) of permutations of (1,..., n) whose order is exactly k, one obtains via Möbius inversion and (), see Wilf [28], that B (n) ∼ A (n), as n →∞ . k k 123 880 A. Cantón et al. 5.2 Entire Functions: Bell Numbers and Idempotent Functions We apply next Theorems 4.1 and C to the egfs of the Bell numbers and of the number of idempotent functions, which are both entire functions. 5.2.1 Bell Numbers e −1 Consider ﬁrst the function f (z) = e , the egf of the Bell numbers B described above in Sect. 2.1.6. Result b) of Hayman in Sect. 3.3.1 already gives that f is in the Hayman class, but let us see now how this fact follows most easily from Theorem 4.1 on our way to obtain the Moser–Wyman asymptotic formula (5.5) below for the Bell numbers, [22]. e −1 t 2 t For f (z) = e , we have that m(t ) = te and that σ (t ) = t (t + 1)e .Inthe notation of Theorem 4.1, the function g is g(z) = e − 1. For a potential cut h, condition (4.3) for the major arc requires that 3 t 3 lim t e h(t ) = 0 . t ↑∞ For the minor arc, using that t cos θ t t 2 e − e ≤− e θ , for any t ≥ 1 and |θ | <π , (5.4) 2π we obtain, for t ≥ 1, that ı θ ı θ te t t cos θ t t 2 Reg(te ) − g(t ) ≤|e |− e = e − e ≤− e θ . 2π And so ı θ t 2 exp sup Reg(te ) − g(t ) ≤ exp − e h(t ) . 2π h(t )≤|θ |≤π −αt With h(t ) = e ,for t > 0 and ﬁxed α ∈ (1/3, 1/2), both conditions (4.3) e −1 and (4.4) are satisﬁed, and we conclude that e is in the Hayman class. The t for f are such that m(t ) = t e = n. Thus t = W (n), where W is the n n n n Lambert function. Since f is strongly Gaussian, formula (3.7) gives that W (n) e −1 B 1 e ∼ √ , as n →∞ , (5.5) W (n) n n! 2π W (n)(W (n) + 1)e W (n) which is the asymptotic formula for the Bell numbers of Moser–Wyman, [22]. 123 Khinchin Families and Hayman Class 881 5.2.2 Counting Idempotent Functions Consider now the function ze n f (z) = e = z , n! n=0 which turns out to be the egf of idempotent functions. That is, U denotes the number of idempotent functions u of {1,..., n} into itself, i.e., functions u such that u ◦ u ≡ u. See [9, p. 571]. We will verify shortly that f is in the Hayman class, towards proving the Harris– Schoenfeld ([14,15]) asymptotic formula (5.6) below for the U . For that aim we could appeal to the combination of results b) and c) of Hayman of Sect. 3.3.1, but we favour a direct proof via Theorem 4.1. z 2 t 2 3 2 t For f (z) = exp(ze ), we have that m(t ) = (t + t )e and σ (t ) = (t + 3t + t )e , for t > 0. The function g is, in this case, g(z) = ze . For a potential cut h towards applying Theorem 4.1, observe that condition (4.3) for the major arc requires that 4 t 3 lim t e h(t ) = 0 . t ↑∞ For the minor arc, using again (5.4), we have, for t > 1, that ı θ ı θ t cos θ ıt sin θ t t cos θ t t 2 Reg(te ) − g(t ) ≤|te e e |− te = te − te ≤− te θ , 2π so that ı θ t 2 exp sup Reg(te ) − g(t ) ≤ exp − te h(t ) , for t > 1 . 2π h(t )≤|θ |≤π −αt As before, with α ∈ (1/3, 1/2) and h(t ) = e ,for t > 0, both (4.3) and (4.4) are fulﬁlled, and we deduce from Theorem 4.1 that f is strongly Gaussian. We now obtain the promised asymptotic formula for the U . Since f is strongly Gaussian, we obtain, with t given by m(t ) = t (t + 1)e = n, for each n ≥ 1, that n n n n U 1 1 n ∼ √ exp , as n →∞ , (5.6) n+3/2 n! t /2 1 + t 2π n e t n the asymptotic formula due to Harris–Schoenfeld [14,15]. 123 882 A. Cantón et al. 5.3 Functions in D:Cover Mapof {z ∈ C :|z| > 1} Consider now 1 + z f (z) = exp = A z , for z ∈ D . 1 − z n=0 This function f is a universal cover of {z ∈ C :|z| > 1}. In this case R = 1, and the function g is given by g(z) = (1 + z)/(1 − z),for z ∈ D. The mean and variance functions of f are given by 2t 2t (1 + t ) m(t ) = and σ (t ) = for t ∈ (0, 1). 2 3 (1 − t ) (1 − t ) The variance condition (4.2) is clearly satisﬁed. Now we look for a cut function h of the form h = (1 − t ) with some appropriate α> 0, towards applying Theorem 4.1. The condition on the major arc (4.3) reduces 3 4 to lim h(t ) /(1 − t ) = 0. For the exponent α, this translates into requiring that t ↑1 α> 4/3. For the minor arc, we start by writing g as g(z) = − 1 . 1 − z If the exponent α in the deﬁnition of h satisﬁes α> 1, then lim h(t )/(1 − t ) = 0, t ↑1 and then Lemmas 4.3 and 4.4 give that (1 − t ) ı θ lim sup sup Re g(te ) − g(t ) h(t ) t ↑1 h(t )≤|θ |≤π (1 − t ) 1 1 ≤ 2 lim − =−1 . 2 ıh(t ) t ↑1 h(t ) 1 − te 1 − t And so, for some t ∈ (0, 1) and δ> 0, we have that h(t ) ı θ sup Re(g(te ) − g(t )) ≤−δ , for t ∈ (t , 1). (1 − t ) h(t )≤|θ |≤π 3/2 Since σ(t ) 1/(1 − t ) ,as t ↑ 1, we deduce that (4.4) is satisﬁed as long as α< 3/2. Thus, with α ∈ (4/3, 3/2) for the cut function, we conclude that f is in the Hayman class. Next, by appealing to the substitution Theorem C, we obtain an asymptotic formula for the Taylor coefﬁcients A of f (z). 123 Khinchin Families and Hayman Class 883 2 3/2 Deﬁne m (t ) = 2/(1 − t ) and σ(t ) = 2/(1 − t ) ,for t ∈ (0, 1), and observe that σ(t ) ∼ σ(t), as t ↑ 1 , and that m (t ) − m(t ) = = o(σ (t )) , as t ↑ 1 , 1 − t and, thus, that condition (3.9) is satisﬁed. For each n ≥ 1, take τ given by 1 − τ = 2/n. Since n n −n τ = exp n ln √ = exp 2n + 1 + o(1) , as n →∞ , 1 − 2/n by virtue of formula (3.10) we deduce that 2 2 n 1 1 e A ∼ √ , as n →∞ . 1/4 3/4 2 n 2π This asymptotic formula is Theorem XIII of Hayman’s [16]. See also Wright [30] and Macintyre–Wilson [19]. For the closely related egf z I exp = z 1 − z n! n=0 of the number I of fragmented permutations of (1,..., n) we have, with a similar z/(1−z) argument, that e is in the Hayman class and that I 1 1 2 n ∼ √ e , as n →∞ . 3/4 n! n 2 e π See [9, p. 563]. Remark 5.2 For the functions 1 1 f (z) = exp γ , for z ∈ D , (1 − z) 1 − z with integers β ≥ 0 and γ ≥ 1, considered by Wright [30] and Macintyre–Wilson in [19] (both with more general β and γ ), Theorem 4.1 gives us that f is strongly Gaussian. The function g(z) is in this case 1 1 g(z) = β ln + γ · 1 − z 1 − z 123 884 A. Cantón et al. For the estimate of the minor arc, just observe that 1 1 ı θ Reg(te ) − g(t ) ≤ γ Re − . ı θ 1 − te 1 − t Appropriate approximations m and σ are γ 2γ m (t ) = and σ (t ) = , 2 3 (1 − t ) (1 − t ) which satisfy condition (3.9). For the coefﬁcients a of f (z) we have 1/4−β/2 γ/2 1 γ e 1 2 γ n a ∼ √ √ e , as n →∞ . 3/4−β/2 2π 2 The functions f (z) = exp(1/(1 − z) ), for integer ρ> 0, also considered by Wright [32] (with more general exponents ρ), are also strongly Gaussian. A cut function h(t ) = (1 − t ) ,for α> 0, satisﬁes the condition on the major arc if 3α> ρ + 3. Using Lemmas 4.3 and 4.4 with k = ρ, we see that h satisﬁes the condition on the minor arc if 2α< 2 + ρ. ρ+1 1/(1+ρ) given by ρt /(1−t ) = n (so that 1−t ∼ (ρ/n) ,as n →∞), we With t n n n n have an asymptotic formula for the coefﬁcients of f , but unless ρ = 1, the substitution Theorem C is of no avail for simplifying the resulting expression. 6 Partitions Next we discuss ogfs f of different sorts of partitions: P for usual partitions, Q for partitions with distinct parts, P for partitions with parts in an arithmetic sequence, a,b 1 b or W , which codiﬁes plane partitions, or W for colored partitions. 1 1 As in Sect. 5, we will use Theorem 4.1 to verify that all these partition functions are strongly Gaussian and then use (3.10) to provide asymptotic formulas for their coefﬁcients. But in sharp contrast to the power series dealt with in Sect. 5, the corresponding functions g do not have closed formulas and the veriﬁcations of the conditions of Theorem 4.1 are more involved. In addition, the mean and variance functions do not have closed formulas either and this requires obtaining suitable approximations of these functions to ﬁnd an approximation τ of t . And, ﬁnally, a proper asymptotic n n formula for the a requires, in this setting, determining the asymptotic behaviour of f on (0, R) in order to be able to get an asymptotic formula for f (t ). The variance condition and the major arc requirement of Theorem 4.1 are quite direct; it is the minor arc condition which requires more work. We will obtain the needed approximations of means and variances in Sect. 6.1, the asymptotic expressions on (0, 1) in Sect. 6.2 and, ﬁnally, we will discuss strong gaussianity and asymptotic formulae of their coefﬁcients in Sect. 6.3. 123 Khinchin Families and Hayman Class 885 These asymptotic formulae of coefﬁcients a of partition functions turn out to be always of the form 1 δ γ n a ∼ α e , as n →∞ , for appropriate α, β, γ , δ > 0. 6.1 Approximation of Means and Variances We collect here convenient approximations of the mean and variance functions, satis- fying in each case condition (3.9) of Theorem C, of the Khinchin families of a number of partition functions. Consider ﬁrst the partition function P(z) = , for |z| < 1 , 1 − z j =1 with Khinchin family ( X ) and mean and variance functions t t ∈[0,1) ∞ ∞ j 2 j jt j t m(t ) = and σ (t ) = , for t ∈ (0, 1). j j 2 1 − t (1 − t ) j =1 j =1 The precise values of the integrals in the following lemma will be invoked frequently in the rest of the paper. Lemma 6.1 a) s ln ds = ζ(u + 2)(u + 1), for u > −1 , −s 1 − e ∞ −s b) s ds = ζ(u + 1)(u + 1), for u > 0 , −s 1 − e ∞ −s c) s ds = ζ(u)(u + 1), for u > 1 . −s 2 (1 − e ) −s It is convenient to consider m(e ),for s > 0, and then s ↓ 0. We have − js 1 e −s m(e ) = js · − js s (1 − e ) j =1 Let φ be the function −x xe φ(x ) = , for x > 0 . −x 1 − e 123 886 A. Cantón et al. If we set φ(0) = 1, then φ ∈ C [0, +∞) and φ(s)ds = ζ(2),byb)ofLemma 6.1. In addition, we have that lim φ(t ) = 0 and |φ (s)|ds < +∞.Fromthe Euler t →∞ summation formula (1.1) applied to φ, we get that ζ(2) −s sm(e ) − = O(1), as s ↓ 0 . (6.1) 2 −s and, therefore, that lim s m(e ) = ζ(2), and so s↓0 ζ(2) −s m(e ) ∼ , as s ↓ 0 . (6.2) −s 2 If we deﬁne m (e ) = ζ(2)/s ,for s > 0, we have that m(t ) ∼ m (t ),as t ↑ 1. For the variance function σ (t ) we obtain analogously, using Lemma 6.1, that (3)ζ (2) 2 −s σ (e ) ∼ , as s ↓ 0 . (6.3) 2 −s 2 3 Moreover, if we deﬁne σ (e ) = π /(3s ),for s > 0, we have that σ(t ) ∼ σ(t), as t ↑ 1 , and also, using (6.1), that −s −s m(e ) − m (e ) = O( s) = o(1), as s ↓ 0 . −s σ(e ) Next, we turn to Q, the ogf of partitions with distinct parts. For the mean and variance functions of its Khinchin family, using Lemma 6.1 again, plus the identity ln(1 + x ) = ln(1 − x ) − ln(1 − x ) for x ∈ (0, 1), and its ﬁrst two derivatives, we obtain that, as s ↓ 0, − js e ζ(2) −s −s m(e ) = j ∼ m (e ), − js 2 1 + e 2s j =1 (6.4) − js e ζ(2) 2 −s 2 2 −s σ (e ) = j ∼ σ (e ); − js 2 3 (1 + e ) s j =1 and, also, that −s −s m(e ) − m (e ) √ = O( s) = o(1), as s ↓ 0 . −s σ(e ) 123 Khinchin Families and Hayman Class 887 For the ogf of P with parts in the arithmetic progression {aj + b; j ≥ 0}, with a,b integers a, b ≥ 1, we have analogously that, as s ↓ 0, −(aj +b)s e ζ(2) −s −s m(e ) = (aj + b) ∼ m (e ), −(aj +b)s 2 1 − e as j =0 (6.5) −(aj +b)s e 2ζ(2) 2 −s 2 2 −s σ (e ) = (aj + b) ∼ σ (e ); −(aj +b)s 2 3 (1 − e ) as j =0 and, also, −s −s m(e ) − m (e ) √ = O( s) = o(1), as s ↓ 0 . −s σ(e ) For integers a ≥ 1 and b ≥ 0, consider the inﬁnite product W (z) = , for z ∈ D . 1 − z j =1 Let m(t ) and σ(t ) be the mean and variance functions of the Khinchin family of W . Consider the real function a −x x e φ(x ) = x , for x ≥ 0 , −x 1 − e whose derivatives of order less than b vanish at x = 0 and x =+∞, and is such that 1 b + 1 b + 1 φ(x ) dx = ζ 1 + 1 + . a a a Arguing as above for the case of regular partitions, using φ and Euler’s summation of order b + 1, we get that −s −s m(e ) = m (e ) + O , as s ↓ 0 , where 1 b + 1 b + 1 1 −s m (e ) = ζ 1 + 1 + · (6.6) 1+(b+1)/a a a a s Analogously, 1 b + 1 b + 1 1 2 −s 2 −s σ (e ) ∼ ζ 1 + 2 + σ (e ), as s ↓ 0 . 2+(b+1)/a a a a s (6.7) 123 888 A. Cantón et al. Observe that −s −s m(e ) − m (e ) (b+1)/2a = O(s ) = o(1), as s ↓ 0 . (6.8) −s σ(e ) 6.2 Asymptotics of Partition Functions on (0, 1) We start with the ogf of partitions P; an instance we will built upon. For the partition function P we have the well-known asymptotic formula in the interval (0, 1): ζ(2) 1 −s ln P(e ) = + ln s − ln 2π + o(1), as s ↓ 0 , s 2 and, thus, −s ζ(2)/s P(e ) ∼ √ se , as s ↓ 0 . (6.9) 2π See Hardy–Ramanujan [13, Sect. 3.2] (the formula for g(x ) there should have a factor 3/2 (1 − x ) instead of 1 − x). See also de Bruijn [8, Ch. 3, Ex. 3] or even Báez-Duarte [2, (2.21)]. We provide a proof since its ingredients are to be used below to handle other partition functions. Fix s > 0 and apply Euler’s summation of order 2 to the function f (x ) =− ln(1 − −sx e ). Observe that lim f (x ) = lim f (x ) = 0. Write x →∞ x →∞ ∞ ∞ 1 1 B ({x }) −s ln P(e ) = f ( j ) = f (x )dx + f (1) − f (1) − f (x ) dx . 2 12 2! 1 1 j =1 where B ( y) is the second Bernoulli polynomial. Now, using a) in Lemma 6.1 we have that ∞ ∞ 1 1 1 ln dx = ln dx −sx −x 1 − e s 1 − e 1 s ∞ s 1 1 1 1 = ln dx − ln dx −x −x s 1 − e s 1 − e 0 0 ζ(2) = + ln s − 1 + O(s), as s ↓ 0 . Also, 1 1 1 1 f (1) − f (1) =− ln s + + O(s), as s ↓ 0, 2 12 2 12 and ∞ ∞ 2 sx B ({x }) (sx ) e B ({x }) 1 2 2 f (x ) dx = dx . sx 2 2 2! (e − 1) 2! x 1 1 123 Khinchin Families and Hayman Class 889 2 y y 2 Since the function y → ( y e )/(e − 1) is bounded in [0, +∞) and tends to 1 as y ↓ 0, dominated convergence gives that ∞ ∞ B ({x }) B ({x }) 1 2 2 f (x ) dx = dx + o(1), as s ↓ 0 . 2! 2! x 1 1 Thus, ζ(2) 1 1 B ({x }) 1 −s ln P(e ) = + ln s − 1 + − dx + o(1), as s ↓ 0 . s 2 12 2! x We may identify the constant term of the above expression by using analogous Euler’s summation of order 2 between 1 and N for the function ln x, to obtain Stirling’s approximation in the following precise (and standard) form 1 1 1 B ({x }) 1 ln N!= N ln N − N + 1 + ln N + − 1 + dx , 2 12 N 2! x and thus deduce that 1 B ({x }) 1 1 − + dx = ln 2π. 12 2! x The same argument gives for the ogf Q of partitions into distinct parts that 1 ζ(2) −s Q(e ) ∼ √ exp , as s ↓ 0 . (6.10) 2s and for the ogf P of partitions with parts in the arithmetic progression {aj +b; j ≥ 0} a,b with integers a, b ≥ 1, that 1 b ζ(2) −s b/a−1/2 P (e ) ∼ √ (as) exp , as s ↓ 0 . (6.11) a,b a as 2π Recall that P ≡ P and P ≡ Q. The asymptotic formula for Q may be obtained 1,1 2,1 from the identity (2.9) and the corresponding asymptotic formula (6.9)for P. Now we turn to the ogf W , with integers a ≥ 1 and b ≥ 0. Fix s > 0. We will apply Euler’s summation of order b + 2 to the function f (x ) = x ln , for x > 0 . −sx 1 − e Note that b −s ln W (e ) = f ( j). j =1 123 890 A. Cantón et al. Observe that ∞ ∞ 1 1 1 dy (b+1)/a f (x )dx = y ln − y (b+1)/a a s 1 − e y 1 s 1 1 1 dy (b+1)/a = y ln (b+1)/a − y a s 1 − e y 1 1 1 dy (b+1)/a − y ln (b+1)/a − y a s 1 − e y 1 b + 1 b + 1 1 = ζ 1 + (b+1)/a a a a s 1 1 a − ln − + O(s). b + 1 s (b + 1) ( j ) For the function f at ∞, we have that lim f (x ) = 0, for j ≥ 0. x →∞ We need the values of f and its derivatives up to order b + 1at x = 1. We have 1 1 f (1) = ln = ln + O(s). −s 1 − e s Write 1/a f (x ) = g(s x ), b/a where 1 y b b+a b b g( y) = y ln = y + y ln − a ( y ln y) . a a − y y ! "# $ 1 − e e − 1 ! "# $ =η( y) =ω( y) Observe that ( j ) ( j ) 1/a f (1) = g (s ), for j ≥ 1. (b− j )/a The function ω is holomorphic near 0, and its Taylor expansion starts with b+a (1/2) y , while ⎛ ⎞ b! 1 ( j ) b− j ⎝ ⎠ η ( y) = y ln y + , for 1 ≤ j ≤ b and y > 0 . (b − j )! b + 1 − i i =1 (b+1) (b+2) 2 Also, η ( y) = b!/ y and η ( y) =−b!/ y ,for y > 0. And, also, (b+1) 1/a (b+1) 1/a 1/a (b+1) 1/a 1/a (b+1) 1/a f (1) = s g (s ) = s ω (s ) − as η (s ) =−b! a + O(s). 123 Khinchin Families and Hayman Class 891 With all this in mind, we have that b+1 1 B j +1 j +1 ( j ) ( j ) ( f (1) + f (∞)) + (−1) ( f (∞) − f (1)) 2 ( j + 1)! j =1 ⎛ ⎞ 1 1 B b! j +1 j +1 ⎝ ⎠ = ln + (−1) ln s 2 s ( j + 1)! (b − j )! j =1 b j B b! 1 B j +1 b+2 j +1 b+2 + a (−1) +a(−1) b!+ O(s). ( j + 1)! (b − j )! b + 1 − i (b + 2)! j =1 i =1 Applying Euler’s summation of order b + 1to x , with b ≥ 1, just in the inter- val [0, 1], and taking into account that its bth derivative is a constant (b!, actually), we deduce that b−1 1 1 b! j +1 j +1 1 = + + (−1) . b + 1 2 ( j + 1)! (b − j )! j =1 Thus, B b! B 1 1 j +1 b+1 j +1 b+1 (−1) = (−1) + − , ( j + 1)! (b − j )! b + 1 2 b + 1 j =1 for b ≥ 1, and for b = 0also. So, we may simplify and write b+1 1 B j +1 j +1 ( j ) ( j ) ( f (1) + f (∞)) + (−1) ( f (∞) − f (1)) 2 ( j + 1)! j =1 1 B b+1 b+1 = − + (−1) ln s b + 1 b + 1 b j B b! 1 B j +1 b+2 j +1 b+2 + a (−1) + a(−1) b!+ O(s). ( j + 1)! (b − j )! b + 1 − i (b + 2)! j =1 i =1 Finally, B ({x }) b+2 b+3 (b+2) (−1) f (x ) dx (b+2)! B ({x }) dx b+2 b+3 1/a 2 (b+2) 1/a = (−1) (s x ) g (s x ) . (b+2)! x 123 892 A. Cantón et al. 2 (b+2) The function y → y g ( y) is bounded in (0, ∞) and as y ↓ 0 tends to b! a.We deduce from dominated convergence that, as s ↓ 0, ∞ ∞ B ({x }) B ({x }) dx b+2 b+2 b+3 (b+2) b+3 (−1) f (x ) dx = (−1) b! a + o(1). (b + 2)! (b + 2)! x 1 1 From this, we have 1 b + 1 b + 1 1 B b+1 b −s b+1 ln W (e ) = ζ 1 + + (−1) ln s (b+1)/a a a a s b + 1 b j a B b! 1 j +1 b+1 − + a (−1) (b + 1) ( j + 1)! (b − j )! b + 1 − i j =1 i =1 B B ({x }) dx b+2 b+2 b+2 b+3 + a(−1) b!+ (−1) b! a + o(1). (b + 2)! (b + 2)! x The constant term of the above expression may be written more compactly by appealing to the so called (generalized) Glaisher–Kinkelin constants appearing in (in fact, deﬁned by) the asymptotic formula of (generalized) hyperfactorials. If we apply Euler’s summation of order b + 2 to the function x ln x between 1 and N , we obtain that 1 1 1 b b+1 b+1 b n ln n = N ln N − N + N ln N b + 1 (b + 1) 2 n=1 B b! j +1 j +1 b− j + (−1) N ln N ( j + 1)! (b − j )! j =1 b−1 B b! 1 j +1 j +1 b− j (‡) + (−1) N ( j + 1)! (b − j )! b − i + 1 j =1 i =1 B 1 B b! 1 b+1 j +1 b+1 j +1 + (−1) H + − (−1) b + 1 (b + 1) ( j + 1)! (b − j )! b − i + 1 j =1 i =1 B b! dx 1 b+2 b+2 b+3 − (−1) b!− (−1) B ({x }) + O . b+2 (b + 2)! (b + 2)! x N Here, H denotes the bth harmonic number. This expression (‡) is the analogue of Stirling’s formula but now for the hyper- factorials j of order b; the factorials being the case b = 0. The constant j =1 term (the sum of the terms not depending on N ) in the above expression is ln GK . These GK , b ≥ 1 are the generalized Glaisher–Kinkelin constants introduced by Bendersky, [5], and identiﬁed in the following closed form: b+1 (−1) B b+1 () ln GK = H − ζ (−b), for b ≥ 1 , b b b + 1 123 Khinchin Families and Hayman Class 893 by Choudhury [7], and Adamchik [1]. See also Wang [27]. The proper (original) Glaisher–Kinkelin constant is GK . By taking H = 0, formula () is also valid for 1 0 b = 0. The constant GK is actually 2π. With this notation, we have that, as s ↓ 0, 1 b + 1 b + 1 1 b −s ln W (e ) = ζ 1 + −ζ(−b) ln s+a ζ (−b)+o(1), (b+1)/a a a a s (6.12) where for compactness we have used that ζ(−b) = (−1) B /(b + 1),for b ≥ 0. b+1 6.3 Hayman Class and Coefficients of Partition Functions Let us apply Theorem 4.1 to check that these partition functions, P, Q, P and a,b W , are in the Hayman class and, thus, that they are strongly Gaussian. The variance and major arc conditions of Theorem 4.1 are obtained quite directly in each case. As usual, the minor arc estimates demand more attention. Recall that in Sect. 6.1 we have obtained, for these partition functions, suitable approximations of their mean and variance functions all satisfying condition (3.9) of Theorem C, and that we have just described in Sect. 6.2 appropriate asymptotics of these partition functions in the interval (0, 1). 6.3.1 Partition Function P. Theorem of Hardy–Ramanujan We start, as usual, with the case of the partition function P. Theorem 6.2 The partition function P is in the Hayman class. Proof We check that P satisﬁes the conditions of Theorem 4.1. For the variance function we have, see (6.3), that (3)ζ (2) 2 −s () σ (e ) ∼ , as s ↓ 0 . Thus, the variance condition of Theorem 4.1 is satisﬁed. g(z) We have that P(z) = e ,for z ∈ D, where g is the function ∞ ∞ jk k 1 z 1 z g(z) = ln = = , for z ∈ D j k 1 − z k k 1 − z j =1 j ,k≥1 k=1 (g has non-negative Taylor coefﬁcients, as the second expression above shows). Next we search for a cut function h so that conditions (4.3) and (4.4) of Theorem 4.1 are satisﬁed. We will take h(t ) = (1 − t ) ,for t ∈ (0, 1), for an appropriate α> 0to be speciﬁed shortly. Let F be the auxiliary function of P.Now, −s t g (t ) ≤ F (−s), for t = e and s > 0 , 123 894 A. Cantón et al. and so, because of the asymptotic formula (3.3), we have that t g (t ) = O , as t ↑ 1 . (1 − t ) Thus, if α> 4/3, the cut function h satisﬁes the major arc condition (4.3). ı θ For the minor arc, write for z = te , with t ∈ (0, 1) and θ ∈ R k k 1 z t ı θ Re(g(te )) − g(t ) = Re − . k k k 1 − z 1 − t k=1 All the summands in the series above are non- positive, so that keeping only the term corresponding to k = 1, we deduce that z t 1 1 ı θ Re(g(te )) − g(t ) ≤ Re − = Re − · 1 − z 1 − t 1 − z 1 − t Because of Lemma 4.3, we have that 1 1 1 1 ı θ sup Re(g(te )) − g(t ) ≤ Re − ≤ − · ıh(t ) ıh(t ) 1 − te 1 − t 1 − te 1 − t h(t )≤|θ |≤π And thus, from Lemma 4.4 we deduce, for some δ> 0 and some t ∈ (0, 1), that h(t ) ı θ sup Re(g(te )) − g(t ) ≤−δ , for t < t < 1 . (1 − t ) h(t )≤|θ |≤π Finally, from the moderate growth of σ(t ) given by (), we conclude that condi- tion (4.4) on the minor arc is amply satisﬁed as long as α< 3/2. In conclusion, by taking α so that 4/3 <α < 3/2, all the conditions of Theorem 4.1 are satisﬁed, and P is in the Hayman class. We can now deduce the asymptotic formula for p(n), the coefﬁcients of the partition −s 2 function P.From (6.2) and (6.3) we know that we can take m (e ) = ζ(2)/s and 2 −s 3 σ (e ) = (3)ζ (2)/s satisfying (3.9). −s Thus with τ = e and s = ζ(2)/n,for n ≥ 1, by appealing to formula (3.10) n n and the asymptotics (6.9)of P on (0, 1), we deduce that √ √ 1 1 π 2/3 n p(n) ∼ e , as n →∞ ; (6.13) 4 3 the Hardy–Ramanujan partition theorem, [13]. 123 Khinchin Families and Hayman Class 895 6.3.2 Function Q For the ogf Q of partitions with distinct parts given by ∞ ∞ ∞ n j Q(z) = q(n)z = (1 + z ) = , for z ∈ D , 2 j +1 1 − z n=0 j =1 j =0 we have Q = exp(g), where k+1 (−1) 1 kj k(2 j +1) g(z) = z = z , for z ∈ D . k k k, j ≥1 k≥1; j ≥0 The second expression shows that g has non-negative coefﬁcients. As in the case of P,wehave g (t ) = O(1/(1 − t ) ),as t ↑ 1. Thus, for a cut h(t ) = (1 − t ) , we just need α> 4/3 to fulﬁll condition (4.3) for the major arc of Theorem 4.1. ı θ Now, the minor arc. For z = te with t ∈ (0, 1) and θ ∈ R we have that ı θ 1 + te 2t (cos θ − 1) t (t /2(cos θ −1) (Rez−|z|)/2 = 1 + ≤ 1 + (cos θ − 1) ≤ e = e , 1 + t (1 + t ) 2 and, so, | Q(z)| 1 z t 1 1 1 ≤ exp Re − = exp Re − , Q(|z|) 4 1 − z 1 − t 4 1 − z 1 − t and, consequently, | Q(z)| 1 1 1 Reg(z) − g(|z|) = ln ≤ Re − . Q(|z|) 4 1 − z 1 − t With this and the same argument as in the case of P, we see that the cut function h(t ) = (1 − t ) with α< 3/2 satisﬁes the condition for the minor arc of Theorem 4.1. Thus, with α such that 4/3 <α < 3/2, we conclude that: Theorem 6.3 The partition function Q (of partitions with distinct parts) is in the Hayman class. The approximations m and σ for the mean and variance functions of the Khinchin family of Q given in formula (6.4), and the asymptotics (6.10)of Q in the interval (0, 1), plus strong gaussianity and Theorem C, allow us to derive the asymptotic formula for the number q(n) of partitions of n with distinct parts, see [13]: √ √ 1 1 π 1/3 n q(n) ∼ e , as n →∞ . 1/4 3/4 4 · 3 n 123 896 A. Cantón et al. 6.3.3 Function P : Theorem of Ingham a,b We consider now, for integers a, b ≥ 1, the ogf P (z) = , for z ∈ D , a,b aj +b 1 − z j =0 of partitions with parts in the arithmetic progression {aj + b : j ≥ 0}. If gcd(a, b) = d > 1, then P is not strongly Gaussian (although it is Gaussian), a,b since we can write P (z) = G(z ), where G is a holomorphic function in D, and a,b thus the coefﬁcients of P satisfy no asymptotic formula; see Remark 6.6 below. a,b We will verify by means of Theorem 4.1 that if gcd(a, b) = 1, then P is strongly a,b Gaussian. The condition that a and b are relatively prime is used exclusively to verify the minor arc condition of Theorem 4.1. We have already obtained, see formula (6.5), convenient approximations m and σ of the mean and variance functions of the Khinchin family of P , and an appropriate a,b asymptotic formula (6.11)of P on the interval (0, 1). a,b We can write P = exp(g), where a,b ∞ ∞ bk 1 1 z 1 k(aj +b) k g(z) = z = = U (z ), for z ∈ D , ak k k 1 − z k k=1 k=1 k≥1; j ≥0 where U is the rational function U (z) = , 1 − z holomorphic in D. Observe that the Taylor coefﬁcients of U around z = 0 are non- negative. We search now for a cut function h(t ) = (1 − t ) with appropriate α> 0. Since g (t ) = O(1/(1 − t ) ),as t ↑ 1, the condition (4.3) of Theorem 4.1 for the major arc is satisﬁed if α> 4/3. Next, the minor arc. This is where we use the condition gcd(a, b) = 1. We have k k Reg(z) − g(|z|) = ReU (z ) − U (|z| ) ≤ ReU (z) − U (|z|), for z ∈ D . k=1 The inequality holds since all the summands in the series above are non-positive. 2πıj /a The function U (z) has simple poles at the ath roots of unity: γ = e , with b+1 0 ≤ j < a. The residue of U at γ is −γ /a. ı φ Let us denote by R the region R ={z = re : 1/2 ≤ r < 1, |φ − 2π j /a|≤ j j a−1 π/a}, and let R = R ; observe that we do not include R in R. j 0 j =1 123 Khinchin Families and Hayman Class 897 For a certain constant M > 0 wehavethat b+1 () U (z) + ≤ M , for z ∈ R and 0 ≤ j < a . a(z − γ ) We start analyzing ReU (z) for z ∈ R. Lemma 6.4 For 0 < j < a, we have that 1 −|z| 1 b lim sup sup Re U (z) ≤ 1 + cos 2π j . |z| 2 a t ↑1 z∈R ; t ≤|z|≤1 Moreover, for certain δ ∈ (0, 1) and t ∈ (1/2, 1) we have that 1 −|z| Re U (z) ≤ δ, for every z ∈ R with |z| > t . |z| Proof Fix 0 < j < a and z ∈ R .Let w ∈ R such that z = γ w. By appealing j 0 j to () we have that a a a 1 −|z| 1 −|z| 1 −|w| 1 −|z| () Re U (z) ≤ Re γ + M . b b b |z| a(1 −|z|)|z| 1 − w |z| Now, for every w ∈ D one has that 1 −|w| 1 1 ∈ D , ∪{1} , 1 − w 2 2 and, in particular, 1 −|w| 1 ı η Re e ≤ (1 + cos(η)) , for all η ∈ R and w ∈ D . 1 − w 2 Therefore, we obtain from () that a a a 1 −|z| 1 −|z| 1 −|z| () Re U (z) ≤ δ + M . b b b |z| a(1 −|z|)|z| |z| with δ = (1 + cos(2πıjb/a)) . This bound () gives the lim sup of the statement. The bound now follows since gcd(a, b) = 1 and 0 < j < a imply that cos(2πıjb/a)< 1. As a consequence of Lemma 6.4,wehave 1 − δ () Re U (z) − U (t ) ≤ (δ − 1)U (t ) ≤− , for z ∈ R with |z|= t ∈ (t , 1). a 1 − t 123 898 A. Cantón et al. It remains to analyze ReU (z) in the region R .For z ∈ R , we have that 0 0 U (z) − ≤ M , a(1 − z) and, therefore, for z ∈ R with |z|= t, we have that 1 1 1 1 t Re U (z) − U (t ) ≤ M + Re − + − · a 1 − z 1 − t a(1 − t ) (1 − t ) Now, 1 t b − ≤ , for each t ∈ (0, 1), a(1 − t ) (1 − t ) a and thus from Lemmas 4.3 and 4.4, arguing as in the case above of P and Q, and increasing the t above if necessary, we deduce that for certain δ> 0, b h(t ) () sup (Re U (z) − U (t )) ≤ M + − δ , for t ∈ (t , 1). ı θ a (1 − t ) z=te ∈R ; h(t )≤|θ | The minor arc for {|z|= t } comprises R ∩{|z|= t } (where we have the bound ()) ı θ and the two subarcs in R ∩{|z|= t } of those z = te where |θ|≥ h(t ). From () and () and the moderate growth of σ,see (6.5), we conclude thatα< 3/2 sufﬁces to guarantee the minor arc condition of Theorem 4.1, and consequently that: Theorem 6.5 If gcd(a, b) = 1,P is in the Hayman class. a,b The approximations m and σ for the mean and variance functions of the Khinchin family of P given in (6.5) and the asymptotics (6.11)of P in the interval (0, 1), a,b a,b plus strong gaussianity and Theorem C, allow us to derive Ingham’s theorem, originally in [17], seealsoKane [18]: for integers a, b ≥ 1 with gcd(a, b) = 1, the number of partitions p (n) of n ≥ 1 in parts drawn for the arithmetic sequence {aj + b : j ≥ 0} a,b satisﬁes that, as n →∞, % & b/(2a) 1 b π 1 2 b/(2a)−1/2 p (n)∼ √ a exp π n . a,b 1/2+b/(2a) a 6 n 3a 22π (6.14) Remark 6.6 If gcd(a, b) = d > 1, then we have P (z) = P (z ), for z ∈ D , a,b a ,b 123 Khinchin Families and Hayman Class 899 where a = a/d and b = b/d. Observe that gcd(a , b ) = 1. We deduce from (6.14) that, as n →∞, coef P = coef P [nd] a,b [n] a ,b % & b/(2a) 1 b π b/(2a)−1/2 ∼ d √ a a 6 22π 1 2 × exp π nd . 1/2+b/(2a) (nd) 3a All the coefﬁcients of P (z) whose indices are not multiples of d are 0. a,b 6.3.4 Plane and Colored Partitions: Wright’s Theorem We deal here with W , where b ≥ 0, the ogf of colored partitions with coloring sequence ( j ) . j ≥1 First we discuss the case W and then we describe the minor changes to approach the general case b ≥ 1. a) Plane partitions. The MacMahon function ∞ ∞ 1 n W (z) = M z 1 − z j =1 n=0 codiﬁes plane partitions: M is the number of plane partitions of the integer n,for n ≥ 1, with M = 1. We have already obtained, see (6.6) and (6.7), convenient approximations, m and σ , of the mean and variance functions of the Khinchin family of W : 1 1 −s 2 −s m (e ) = ζ(3)(3) and σ (e ) = ζ(3)(4) , as s ↓ 0 , 3 4 s s which satisfy condition (3.9). We also have at our disposal, see equation (6.12), a suitable asymptotic formula for W on the interval (0, 1): 1/12 2 e 2 1 −s ζ (−1) 1/12 ζ(3)/s 1/12 ζ(3)/s W (e ) ∼ e s e = s e , as s ↓ 0 . (6.15) GK We apply now Theorem 4.1 to verify that W is strongly Gaussian. The variance condition is clearly satisﬁed. We look now for a cut function h(t ) = (1 − t ) , with α> 0 to be speciﬁed. Let g(z) = ln W (z) in D.Wehave j 1 l j l g(z) = (z ) = K (z ), for z ∈ D , l l l, j ≥1 l=1 123 900 A. Cantón et al. where K is the Koebe function: z 1 1 + z 1 K (z) = = − , for z ∈ D . (1 − z) 4 1 − z 4 Observe, in particular, that the coefﬁcients of g are all non-negative. As g (t ) = O(1/(1 − t ) ),as t ↑ 1, see (3.4), the condition on the major arc (4.3) is readily satisﬁed if α> 5/3. ı θ Now, the minor arc. For z = te with t ∈ (0, 1) and θ ∈ R, we have that l l Reg(z) − g(t ) = Re K (z ) − K (t ) ≤ Re K (z) − K (t ) l=1 2 2 1 1 + z 1 + t = Re − , 4 1 − z 1 − z since the summands above are all non-positive, and, thus, that 2 2 1 1 + z 1 + t 1 1 1 Reg(z) − g(t ) ≤ − ≤ (1 + t ) − 2 2 4 1 − z 1 − t 4 |1 − z| (1 − t ) 1 1 1 1 1 1 − t ≤ − = − 1 . 2 2 2 ı θ 4 |1 − z| (1 − t ) 4 (1 − t ) 1 − te Appealing to Lemma 4.4 we deduce that for t ≥ t and some δ> 0 we have that h(t ) sup (Reg(z) − g(t )) ≤−δ , (1 − t ) |θ |≥h(t ) and conclude that condition (4.4) on the minor arc is satisﬁed whenever α< 2. Thus, specifying α ∈ (5/3, 2), we conclude that: Theorem 6.7 The MacMahon function W is in the Hayman class. −s 1/3 For each n ≥ 1, take τ = e with s = (ζ (3)(3)/n) , so that m (τ ) = n. n n n Plugging τ in the asymptotic formula (3.10) and using (6.15) we obtain % & ζ (−1) e 1 7/36 25/36 1/3 2/3 M ∼ √ ζ(3) 2 exp 3(ζ (3)/4) n , as n →∞ , 25/36 12π (6.16) which is Wright’s asymptotic formula for plane partitions, [29]. b) Colored partitions. Fix an integer b ≥ 1. The argument which follows to han- b 1 dle W is just an extension of the one above for plane partitions, W ;ﬁrst,strong 1 1 gaussianity via Theorem 4.1 and then asymptotics of coefﬁcients via Theorem C. Theorem 6.8 For b ≥ 1, the function W is in the Hayman class. 123 Khinchin Families and Hayman Class 901 Proof Introduce the power series Q and R given by b b ∞ ∞ j ! w b j j Q (w) = j w and R (w) = w = b! , b b b+1 ( j − b)! (1 − w) j =1 j =b and observe that ∞ ∞ 1 1 b b kj k g(z) = ln W (z) = j z = Q (z ). k k j ,k k=1 ı θ For z = te ∈ D, we have that Reg(z) − g(t ) ≤ Re Q (z) − Q (t ) b b b b z t 1 1 ≤ b! Re − ≤ b! t − . b+1 b+1 b+1 b+1 (1 − z) (1 − t ) |1 − z| (1 − t ) For a potential cut function h(t ) = (1 − t ) , with α> 0, we have, because of Lemmas 4.3 and 4.4, that for some δ > 0, (b+1)+2 (1 − t ) ı θ lim sup sup Reg(te ) − g(t ) ≤−δ < 0 . h(t ) t ↑1 h(t )≤|θ |≤π Thus, condition (4.4) on the minor arc of Theorem 4.1 is satisﬁed as long as α< 1 + (b + 1)/2. (b+1)+3 From (3.4), we deduce that g (t ) = O(1/(1 − t ) ),as t ↑ 1, and, thus, that condition on the major arc is satisﬁed whenever α> 1 + (b + 1)/3. Now from the approximation of moments of the Khinchin family of W given by (6.6), (6.7) and (6.8), and the asymptotics of W on the interval (0, 1), we deduce from Theorem C the following asymptotic formula for the coefﬁcients a of W : 1 (b+1)/(b+2) γ n a ∼ α e , as n →∞ , (6.17) n b where ζ (−b) (−2ζ(−b)+1)/(2(b+2)) α = √ e √ [(b + 2)ζ(b + 2)] , 2π b + 2 −2ζ(−b) + b + 3 b + 2 1/(b+2) β = ,γ = ((b + 2)ζ(b + 2)) . b b 2(b + 2) b + 1 Formula (6.17) reduces to (6.13) when b = 0, usual partitions, and to (6.16) when b = 1, plane partitions. 123 902 A. Cantón et al. 6.3.5 General W For a ≥ 1, b ≥ 0, general, the ogfs W (z) are all Gaussian. We do have appropriate approximations of the mean and variance functions given by (6.6), (6.7) and (6.8), satisfying condition (3.9) towards applying Theorem C, and also a convenient asymptotic formula in the interval (0, 1) given by (6.12). Besides, aiming at applying Theorem 4.1 and verifying the strong gaussianity b α of W , for a potential cut function h(t ) = (1 − t ) , with α> 0, we do have, because of (3.4), that condition (4.3) on the major arc is satisﬁed as long as α> 1 + (b + 1)/(3a). But, alas, we have not been able to stretch the simple estimates above, for the case a = 1, to verify the minor arc condition (4.4), which should be satisﬁed if α< 1 + (b + 1)/(2a). If that were the case, all the ogfs W would be in the Hayman class, and theefore they would be strongly Gaussian. And from all the above, we would have the following asymptotic formula for the coefﬁcients a of W : 1 (b+1)/(b+1+a) γ n a,b a ∼ α e , as n →∞, n a,b a,b where 1 a (a−2aζ(−b))/(2(a+b+1)) aζ (−b) α = √ e C , a,b a,b a + b + 1 2π −2aζ(−b) + 2a + b + 1 a + b + 1 a/(a+b+1) β = ,γ = C , a,b a,b a,b 2(a + b + 1) b + 1 with 1 b + 1 b + 1 C = ζ 1 + 1 + . a,b a a a For asymptotic formulae (and further asymptotic expansions) in the cases W and W , of partitions into squares and ath powers, obtained via the circle method, we refer to Vaughan [26], Gafni [10] and the primordial Wright [31]. It would be nice to know whether the ogf W , with a ≥ 2, of partitions into powers of a is in the Hayman class. Acknowledgements We are extremely grateful to the referees of an earlier version of this paper for their insightful and valuable suggestions and recommendations. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted 123 Khinchin Families and Hayman Class 903 by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Adamchik, V.S.: Polygamma functions of negative order. J. Comput. Appl. 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Wang, W.: Some asymptotic expansions on hyperfactorial functions and generalized Glaisher–Kinkelin constants. Ramanujan J. 43(3), 513–533 (2017) P(z) 28. Wilf, H.: The asymptotics of e and the number of elements of each order in S . Bull. Am. Math. Soc. (N.S.) 15(2), 228–232 (1986) 29. Wright, E.M.: Asymptotic partition formulae I. Plane partitions. Q. J. Math. 1, 177–189 (1931) 30. Wright, E.M.: The coefﬁcients of a certain power series. J. Lond. Math. Soc. 7(4), 256–262 (1932) 31. Wright, E.M.: Asymptotic partition formulae. III. Partitions into k-th powers. Acta Math. 63, 143–191 (1934) 123 904 A. Cantón et al. 32. Wright, E.M.: On the coefﬁcients of power series having exponential singularities. II. J. Lond. Math. Soc. 24, 304–309 (1949) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. Authors and Aﬃliations 1 2 2 3 Alicia Cantón · José L. Fernández · Pablo Fernández · Víctor J. Maciá José L. Fernández joseluis.fernandez@uam.es Alicia Cantón alicia.canton@upm.es Pablo Fernández pablo.fernandez@uam.es Víctor J. Maciá victor.macia@wustl.edu Departamento de Matemática e Informática Aplicadas a las Ingenierías Civil y Naval, ETSIN, Universidad Politécnica de Madrid, Avenida de la Memoria 4, Ciudad Universitaria, 28040 Madrid, Spain Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco s/n, 28049 Madrid, Spain Department of Mathematics and Statistics, Washington University in St Louis, 1 Brookings Drive, St. Louis, MO 63130, USA http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals http://www.deepdyve.com/lp/springer-journals/khinchin-families-and-hayman-class-23Y4emRC08

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Publisher: Springer Journals
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ISSN: 1617-9447
eISSN: 2195-3724
DOI: 10.1007/s40315-021-00420-6
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Abstract

We give criteria, following Hayman and Báez-Duarte, for non-vanishing functions with non-negative coefﬁcients to be Gaussian and strongly Gaussian. We use these criteria to show in a simple and uniﬁed manner asymptotics for a number of combinatorial objects, and, particularly, for a variety of partition questions like Ingham’s theorem on partitions with parts in an arithmetic sequence, or Wright’s theorem on plane partitions and, of course, Hardy–Ramanujan’s partition theorem. Keywords Khinchin families · Hayman admissible functions · Asymptotic formulae · Partitions · Analytic combinatorics Mathematics Subject Classiﬁcation Primary 30B10; Secondary 05A16 · 11P82 · 60F99 1 Introduction Walter Hayman, in A generalisation of Stirling’s formula,[16], introduced what are now called Hayman (admissible) functions, a concept which has become a cornerstone of asymptotic methods in Analytic Combinatorics. The notion of Hayman function may at ﬁrst glance appear as too speciﬁc and tailor-made for saddle point approximation and for the Laplace method to be applicable. But the Hayman class is a rich collection To Walter K. Hayman, in memoriam Communicated by James K. Langley. Research of J. L. Fernández and P. Fernández supported by Fundación Akusmatika. Research of Victor Maciá partially supported by grant MTM201457769 of Ministerio de Economía y Competitividad of Spain. B José L. Fernández joseluis.fernandez@uam.es Extended author information available on the last page of the article 123 852 A. Cantón et al. of functions, closed under certain operations (exponentials of Hayman functions are Hayman) and small perturbations (the product of a polynomial times a function in the Hayman class returns a function in the Hayman class); its usefulness in Analytic Combinatorics derives mostly from facts like those. A few years later, following an idea of Khinchin, Paul Rosenbloom [25]intro- duced in Complex Analysis a family of probability distributions, which we term here a Khinchin family, associated to any given power series with non-negative coefﬁcients. In this probabilistic framework, he interpreted Hayman functions as power series with Gaussian Khinchin families (see Deﬁnition 3.1) and as a show of its fruitfulness, Rosenbloom proved the Wiman–Valiron theorem on the maximal term of entire func- tions, essentially by applying Chebyshev’s inequality to the corresponding Khinchin family. Luis Báez-Duarte [2] added to this circle of ideas the notion of strongly Gaussian power series (see Deﬁnition 3.4) and a basic substitution theorem (see Theorem C in Sect. 3.2.1) quite useful for asymptotic purposes. Products of power series become, in the Khinchin family side, sums of independent random variables, and by appealing to Lyapunov’s approach to the central limit theorem, Báez-Duarte was able to show j −1 that the function (1 − z ) is strongly Gaussian, and to deduce the classical j =1 Hardy–Ramanujan partition theorem. See also Candelpergher–Miniconi, [6]. In this paper we follow this thread of Hayman, Rosenbloom and Báez-Duarte. We take full advantage of the notion of strong gaussianity and go back to Hayman’s basic function theoretical ideas to extract a direct and purely function theoretical criterion (Theorem 4.1) for belonging to the Hayman class—and so, being strongly Gaussian—for power series which do not vanish in their disk of convergence. We use this criterion, combined with Hayman’s asymptotic formula of Theorem B, to derive in a simple and uniﬁed manner asymptotics for the enumeration of an assort- ment of combinatorial objects. Further, we use Theorem 4.1 to show, with no intervention of the central limit theorem or the circle method, that generating functions of partitions, standard or with a variety of restrictions on parts, are in the Hayman class, and then with the additional and crucial contribution of Theorem C we derive asymptotic formulas for partitions with parts in an arithmetic sequence (Ingham’s theorem), plane partitions (Wright’s theorem), or colored partitions and, of course, Hardy–Ramanujan’s partition theorem. We should emphasize that the general approach of Khinchin families and Hayman class functions aims to give ﬁrst order asymptotic formulae, but not full asymptotic expansions. In forthcoming work, we plan to extend this approach of Khinchin families and apply it to further asymptotic estimations of coefﬁcients like those of inﬁnite products, of large powers of power series, of generating functions of a variety of trees, etc. As it turns out, [16] is one of the most cited papers of Hayman and, as far as we know, he did not pursue its topic any further. The present paper is a natural continuation of [16] once you have at your disposal the Khinchin families framework and the contributions of Báez-Duarte [2]. This paper is dedicated to the memory of Walter Hayman. Maybe, he would have been pleased with the extra range of direct applications, which the present paper contains, of his A generalisation of Stirling’s formula,[16]. 123 Khinchin Families and Hayman Class 853 Plan of the paper. Section 2 expounds the basic theory of Khinchin families. In Sect. 3, the notions of Gaussian and strongly Gaussian power series are introduced, their probabilistic and asymptotic implications, like Hayman’s asymptotic formula, Theorem B, are analyzed, and the Hayman class of functions is presented. In this Sect. 3 we present basic criteria for non-vanishing functions to be Gaussian, Theorem 3.2, and to be strongly Gaussian, Theorem 4.1. This last theorem is applied in Sect. 4 to verify that a number of exponential functions, mostly of combinatorial interest, are strongly Gaussian and to obtain asymptotic formulae for their coefﬁcients. Theorem 4.1 is also the basic tool for the analysis of asymptotic formulae for the number of partitions of various kinds which, with the help of Theorem C, is carried out in Sect. 6. Notations. For positive sequences (a ), (b ), the notation a ∼ b as n →∞ means n n n n that lim a /b = 1; and we refer to that as an asymptotic formula. With a b n→∞ n n n n we abbreviate that c ≤ a /b ≤ C for positive constants c, C. Analogous notations n n are used for positive functions. denotes the distribution function of the standard normal. With ‘ogf’ and ‘egf’ we abbreviate, respectively, ordinary and exponential generating function. The disk of center z ∈ C and radius r > 0 is denoted D(z, r ). The unit disk D(0, 1) is denoted simply by D. E( Z ) and V( Z ) are reserved for the expectation (mean) and variance of the random variable Z.If X and Y are random variables, with X = Y we signify that X and Y have the same distribution. We write {x } for the fractional part of the real number x. The Bernoulli numbers are denoted by B , for j ≥ 0 (note that B =−1/2), while B (x ),for j ≥ 0, stands for the Bernoulli 1 j polynomials. For f ∈ C [0, N ], with k ≥ 1, the Euler summation formula of order k reads: f ( j ) = f (t ) dt j =0 k−1 1 1 B j +1 j +1 ( j ) ( j ) + f (0) + f ( N ) + (−1) f ( N ) − f (0) 2 2 ( j + 1)! j =1 B ({t }) k+1 (k) + (−1) f (t ) dt . (1.1) k! We will frequently refer to the comprehensive treatise Analytic Combinatorics [9] by Flajolet and Sedgewick for background on combinatorial issues. 2 Khinchin Families We denote by K the class of non-constant power series f (z) = a z , n=0 123 854 A. Cantón et al. with positive radius of convergence, which have non-negative Taylor coefﬁcients and such that a > 0. These are the power series of interest in this paper. Observe that f (t ) is increasing for t ∈[0, R), and, since f (0)> 0, we have f (t)> 0, for t ∈[0, R). Deﬁnition 2.1 The Khinchin family of a power series f in K with radius of convergence R > 0 is the family of random variables ( X ) with values in {0, 1,...} and with t ∈[0, R) mass functions given by a t P( X = n) = , for each n ≥ 0 and t ∈ (0, R), f (t ) while X ≡ 0. Any Khinchin family is continuous in distribution in [0, R). No hypothesis upon joint distribution of the variables X is considered: families, not processes. 2.1 Basic Properties 2.1.1 Mean and Variance Functions For the mean and the variance of X we reserve the notations m(t ) = E( X ) and t t σ (t ) = V( X ),for t ∈[0, R). In terms of f , the mean and variance of X may be t t written as tf (t ) m(t ) = ,σ (t ) = tm (t), for t ∈[0, R). (2.1) f (t ) For each t ∈ (0, R), the variable X is not a constant, and so σ (t)> 0. Conse- quently, m(t ) is strictly increasing in [0, R), though, in general, σ(t ) is not increasing, like in the case of polynomials. We denote M = lim m(t). (2.2) t ↑ R 2.1.2 The Case M =∞ This case M =∞, quite relevant in what follows, holds except in some exceptional cases, which we now specify. Let f ∈ K with radius of convergence R > 0 and assume that M < ∞. Then na t tf (t ) n=0 () m(t ) = = ≤ M , for t ∈ (0, R). a t f (t ) n=0 For any ﬁxed t ∈ (0, R), we then have upon integration in () that f (t ) t () ln ≤ M ln , for t ∈[t , R). f 0 f (t ) t 0 0 123 Khinchin Families and Hayman Class 855 Let us distinguish now between radius of convergence R being ﬁnite or not. • If R < +∞ and M < ∞, then () implies that lim f (t)< ∞ and then f t ↑ R a R < ∞. And, besides, from () we deduce also that n=0 () na R < ∞ . n=0 And, conversely, if () holds and f in K has radius of convergence R, then na R n=0 M = < ∞ . a R n=0 • If R =∞ and M < ∞, then () implies that f is a polynomial. And conversely, for a polynomial f in K of degree N , one actually has M = N . In summary, Lemma 2.2 For f (z) = a z in K with radius of convergence R > 0, we have n=0 M < ∞ if and only if R < ∞ and na R < ∞,orif R =∞ and f is a f n n=0 polynomial. ∞ ∞ n n In the ﬁrst case, we have M = ( na R )/( a R ). For a polynomial f n n n=0 n=0 f ∈ K, we have M = deg( f ). In all forthcoming applications, the exceptional cases above will not appear and we will always have M =+∞. Important notation: If M =∞, we denote by t the unique value t ∈[0, R) such f n that m(t ) = n. Observe that the t verify that lim t = R. n n→∞ n 2.1.3 Normalization and Characteristic Functions For each t ∈ (0, R), we denote the normalization of X by X : t t X − m(t ) X = for t ∈ (0, R). σ(t ) The characteristic function of X may be written in terms of the power series f as ∞ ∞ ı θ X ınθ n ınθ E(e ) = e P( X = n) = a t e t n f (t ) n=0 n=0 (2.3) ı θ f (te ) = , for t ∈ (0, R) and θ ∈ R , f (t ) while for its normalized version X we have ı θ X ıθ( X −m(t ))/σ (t ) t t E(e ) = E(e ) (2.4) ı θ X /σ (t ) −ı θ m(t )/σ (t ) = E(e ) e , for t ∈ (0, R) and θ ∈ R , 123 856 A. Cantón et al. and so, ı θ X ı θ X /σ (t ) t t E(e ) = E(e ) , for t ∈ (0, R) and θ ∈ R . 2.1.4 Scale Let f ∈ K have radius of convergence R and Khinchin family ( X ) .Fix an t t ∈[0, R) integer m ≥ 1 and let g (z) be the power series deﬁned by g (z) = f (z ). Then g m m m 1/m is also in K and g has radius of convergence R . Let (Y ) 1/m be the Khinchin family of g .Wehave t m t ∈[0, R ) 1/m Y = mX , for 0 ≤ t < R . t t ∞ n ∞ km For, if f (z) = a z , then g (z) = a z and for each k ≥ 0 and each n m k n=0 k=0 1/m 0 ≤ t < R , we have that km km a t a t k k P(Y /m = k) = P(Y = km) = = = P( X = k). t t t g (t ) f (t ) 2.1.5 Auxiliary Function F The function f does not vanish on the real interval [0, R). And so, it does not vanish in some simply connected region containing that interval. There we may consider ln f , a branch of the logarithm of f which is real on [0, R), and deﬁne the function F (z) = ln f (e ), which is holomorphic in a region containing (−∞, ln R). If f does not vanish anywhere in the disk D(0, R), then the auxiliary function F (z) is deﬁned in the whole half plane Rez < ln R. In general, the auxiliary function F (z) is deﬁned and holomorphic in the half band-like region ={s + ı θ : s < ln R, |θ | < 2/σ (e )}. This follows, for instance, see [3, Prop. 7.8], from the following lemma. ı θ Y 2 Lemma 2.3 If Y is a random variable and E(e ) = 0, then θ V(Y ) ≥ 2. ı θ Combining this lemma and formula (2.3), we deduce that f (te ) = 0, if |θ | < √ √ s+ı θ s z 2/σ (t ), and so f (e ) = 0, if |θ | < 2/σ (e ),for s < ln R. Therefore, f (e ) vanishes nowhere in . Proof of Lemma 2.3 By considering Y − E(Y ), we may assume that E(Y ) = 0. For a complex valued C function h deﬁned in R, we have that y u h( y) − h(0) − h (0) y = h (v) dvdu . 0 0 123 Khinchin Families and Hayman Class 857 ı φ y Applying this identity to h( y) = e , we see that the inequality 2 2 φ y ı φ y |e − 1 − ıyφ|≤ holds for y,φ ∈ R. Substituting Y for y, and θ for φ and taking expectations, we 2 2 2 conclude that 1 ≤ θ E(Y )/2 = θ V(Y )/2. In terms of the auxiliary function F, the mean and variance functions of the Khinchin family of f , given by (2.1), may be expressed as follows: m(t ) = F (s) and σ (t ) = F (s), fort = e and s < ln R . (2.5) 2.1.6 Some Basic Examples We now exhibit explicit formulas for the mean and variance functions and for the characteristic functions of some basic examples. (a) Let f (z) = 1 + z. In this case R =∞, and the mean and variance functions, 2 2 given by (2.1), are m(t ) = t /(1 + t ) and σ (t ) = t /(1 + t ) . For each t > 0, the variable X is a Bernoulli variable with parameter t /(1 + t ), and its characteristic function is, see formula (2.3), ı θ ı θ f (te ) 1 + te ı θ X E(e ) = = , for θ ∈ R and t > 0 , f (t ) 1 + t and thus, using formula (2.4), √ √ ıθ/ t −ı θ t te + e ı θ X E(e ) = , for θ ∈ R and t > 0 . (2.6) 1 + t (b) Let f (z) = 1/(1 − z). In this case R = 1, and the mean and variance functions 2 2 are m(t ) = t /(1 − t ) and σ (t ) = t /(1 − t ) . For each t ∈ (0, 1), the variable X is a geometric variable (number of failures until ﬁrst success) of parameter 1 − t, and its characteristic function is, see formula (2.3), ı θ f (te ) 1 − t ı θ X E(e ) = = , for θ ∈ R and t ∈ (0, 1), ı θ f (t ) 1 − te and thus, using formula (2.4), we have that 1 − t ı θ X −ı θ t E(e ) = √ e ıθ(1−t )/ t 1 − te 1 − t = √ √ , for θ ∈ R and t ∈ (0, 1). (2.7) ı θ t ıθ/ t e − te 123 858 A. Cantón et al. (c) Let f (z) = e . In this case R =∞, and the mean and variance functions are m(t ) = t and σ (t ) = t. For each t > 0, the variable X in its Khinchin family follows a Poisson distribution with parameter t, and its characteristic function is, see formula (2.3), ı θ f (te ) ı θ ı θ X t (e −1) E(e ) = = e , for θ ∈ R and t > 0 , f (t ) and thus, using formula (2.4), we have that ı θ ı θ X ı (θ / t ) E(e ) = exp t e − 1 − √ , for θ ∈ R and t > 0 · (2.8) (d) Let z n f (z) = exp(e − 1) z , for z ∈ C , n! n=0 where B (not to be confused with the nth Bernoulli number B )isthe nth Bell n n number, which counts the number of partitions of the set {1,..., n};see [9, p. 109]. t 2 Then R =∞, and the mean and variance functions are m(t ) = te and σ (t ) = t (t + 1)e . The characteristic function is given by, ı θ f (te ) ı θ ı θ X te t E(e ) = = exp e − e , for θ ∈ R and t > 0 , f (t ) and thus, using formula (2.4), we have that −t /2 ı θ e / t (t +1) t ı θ X te t t /2 E(e ) = exp e − e − ı θ e , for θ ∈ R and t > 0 . t + 1 The asymptotic formula for the B , due to Moser and Wyman [22], is dealt with in Sect. 5.2. 2.1.7 Partition Functions We next turn our attention to generating functions of partition of integers, standard and with a variety of restrictions on the admitted parts. These examples of functions in K are quite relevant in this paper. Asymptotic formulas of their coefﬁcients are dealt with in Sect. 6. The mean and variance functions or the characteristic function of the Khinchin family of these partitions functions are not as direct as in the previous examples. Closed formulas of their mean and variance functions involving series are presented in Sect. 2.2.1, and convenient approximations are exhibited in Sect. 2.2.1. 123 Khinchin Families and Hayman Class 859 (e) The ogf of partitions, the partition function, given by ∞ ∞ P(z) = = p(n) z , for z ∈ D , 1 − z j =1 n=0 is in K. The ogf Q(z) of partitions into distinct parts (which is also the ogf of partitions into odd parts), ∞ ∞ ∞ j n Q(z) = (1 + z ) = = q(n) z , for z ∈ D , 2 j +1 1 − z j =1 j =0 n=0 is also in K. Observe that P(z) Q(z) = , for z ∈ D . (2.9) P(z ) For integers a ≥ 1, b ≥ 1, the inﬁnite product P (z) = , for z ∈ D , a,b aj +b 1 − z j =0 the ogf of the partitions whose parts lie in the arithmetic progression {aj + b : j ≥ 0},isalsoin K. Observe that P ≡ P and that P ≡ Q. 1,1 2,1 (f) For integers a ≥ 1, b ≥ 0, the inﬁnite product W (z) given by W (z) = , for z ∈ D, 1 − z j =1 0 1 is also in K.Wehave W ≡ P.Also, W (z), known as the MacMahon func- 1 1 tion [20], turns out to be the ogf of plane partitions;see [9, p. 580], and Bender–Knuth [4], for a simple proof of this fact. Besides, W (z) is the ogf of partitions with parts which are ath powers of positive integers. In general, W (z) is the ogf of partitions with parts which are ath powers of positive b a integers and with number of colors j for part j ,for j ≥ 1. See Remark 2.4 for the terminology. Remark 2.4 (Colored/weighted partitions) Given a sequence (b ) of integers b ≥ j j ≥1 j 0, the inﬁnite product 1 − z j =1 123 860 A. Cantón et al. is the ogf of colored partitions with coloring sequence (b ) . A colored partition j j ≥1 of n with the above coloring sequence is an array of integers n( j , k) ≥ 0 with j ≥ 1 and 1 ≤ k ≤ b , and n = n(k, j ), i.e., of partitions whose part j may j ≥1;1≤k≤b b b come in b different colours. In the case of W , the coloring sequence is b = j ,for j j j ≥ 1. The colored partitions with the above coloring sequence are partitions in which the part j may appear in any of b available colors, with the order of the colors not mattering. These colored partitions are also called weighted partitions; the weights being the b . See Granovsky–Stark [11], and Granovsky et al. [12], where asymptotics about these (quite general) partitions are approached elegantly and systematically via Meinardus’ theorem, [21], and Khinchin families. 2.1.8 Products and Independence If f , g ∈ K, then the product h = f · g is also in K. If both f and g have radius of convergence at least R, then the product h has radius of convergence at least R.Ifthe respective Khinchin families of f , g and h are ( X ) , (Y ) and ( Z ) , t t ∈[0, R) t t ∈[0, R) t t ∈[0, R) then the law of Z is the law of the independent sum X ⊕ Y of the laws of X and Y : t t t t t P( Z = n) = P( X = k) P(Y = n − k), for each n ≥ 0 . t t t k=0 Thus, products of functions of K become, on the Khinchin family side, sums of independent variables. 2.1.9 Convergence Let ( f ) be a sequence of functions in K, all of them with radius of convergence k k≥1 at least R > 0, which converges uniformly on compacts sets of D(0, R) to a function [k] f ∈ K.Let ( X ) and ( X ) be the Khinchin families of f ,for k ≥ 1, t ∈[0, R) t t ∈[0, R) k and, respectively, of f . [k] Then, for each t ∈[0, R), we have that ( X ) converges in distribution to X k≥1 t as k →∞. [k] Also, for any moment, say, of order q ≥ 1, we have lim E(( X ) ) = E( X ), k→∞ t t since if D is the operator D = z(d/dz), then E( X ) = D ( f )(t )/ f (t ). In particular, for each t ∈[0, R), we have that lim m (t ) = m(t ) and that k→∞ k 2 2 2 2 lim σ (t ) = σ (t ), where m (t ), σ (t ) and m(t ), σ (t ) are, respectively, the k→∞ k k k mean and variance functions of f and f . 123 Khinchin Families and Hayman Class 861 2.2 Infinite Products Let ( f ) be a sequence of functions in K all with radius of convergence at least j j ≥1 R > 0. Assume that the inﬁnite product f (z) = f (z) j =1 converges absolutely and uniformly on compact subsets of D(0, R). The product f has non-negative Taylor coefﬁcients. Since f (0) = 0, for each j, we have f (0) = 0, and actually, f (0)> 0. Moreover, from f (t ) f (t ) = , for t ∈[0, R), f (t ) f (t ) j =1 we see that f the inﬁnite product is non-constant. Thus the product function f is in K and its power series has radius of convergence at least R. Let ( X ) be the Khinchin family of f and let m(t ) = E( X ) and σ (t ) = t t ∈(0, R) t V( X ) be its mean and variance functions. For each j ≥ 1, the Khinchin family of f (z) is denoted by X and its mean and variance functions by m (t ) and σ (t ),for j j ,t j t ∈[0, R). Since f (z) converges uniformly as N →∞ on compact subsets of D(0, R), j =1 we have that, for each t ∈ (0, R),the sum X converges in law to X ,as j ,t t j =1 N →∞, and that for the means and variances we have ∞ ∞ 2 2 m(t ) = m (t ) and σ (t ) = σ (t ), for each t ∈[0, R). j =1 j =1 2.2.1 Partition Functions as Inﬁnite Products As an illustration of the above, we consider now partitions and partitions into distinct parts. Consider ﬁrst the partition function P(z) = , for |z| < 1. 1 − z j =1 Let ( X ) be its Khinchin family. And let (Y ) be the Khinchin family of t t ∈[0,1) t t ∈[0,1) 1/(1 − z). Because of scaling properties, see Sect. 2.1.4, we have that X = jY , for t ∈ (0, 1), j =1 123 862 A. Cantón et al. and each Y is a geometric variable with parameter 1 − t . For its mean and variance functions, m(t ) and σ (t ), we have that ∞ ∞ j 2 j jt j t m(t ) = and σ (t ) = , for t ∈ (0, 1). j j 2 1 − t (1 − t ) j =1 j =1 We will obtain closed form approximations of m(t ) and σ (t ) as t ↑ 1 in Sect. 6.1. If ( X ) is the Khinchin family of the ogf t t ∈[0,∞) Q(z) = (1 + z ) j =1 of partitions into distinct parts, we then have, because of scaling properties, see Sect. 2.1.4, that X = jY j , for t > 0 , j =1 j j and each Y j is a Bernoulli variable with parameter t /(1 + t ). For its mean and variance functions we have that ∞ ∞ j 2 j jt j t m(t ) = and σ (t ) = , for t ∈ (0, 1). j j 2 1 + t (1 + t ) j =1 j =1 Analogous expressions for the mean and variance functions of other partition func- tions like P or W will appear later in Sect. 6.1. a,b a,b 2.3 Hayman’s Identity For f (z) = a z ∈ K, Cauchy’s formula for the coefﬁcient a in terms of the n n n=0 characteristic function of its Khinchin family ( X ) reads t t ∈[0, R) f (t ) ı θ X −ı θ n a = E(e ) e dθ, for each t ∈ (0, R) and n ≥ 1 . 2π t |θ |<π In terms of the characteristic function of the normalized variable X , it becomes, for each t ∈ (0, R) and n ≥ 1, f (t ) ı θ X −ıθ(n−m(t ))/σ (t ) a = E(e ) e dθ. 2π t σ(t ) |θ |<π σ (t ) 123 Khinchin Families and Hayman Class 863 If M =∞, we may take for each n ≥ 1 the (unique) radius t ∈ (0, R) so that f n m(t ) = n, to write f (t ) n ˘ ı θ X a = E(e ) dθ, for each n ≥ 1 , (2.10) 2π t σ(t ) |θ |<π σ (t ) n n which we call Hayman’s identity. If we write this identity (2.10)as a 2π t σ(t ) n n ˘ n ı θ X = E(e ) dθ f (t ) n |θ |<π σ (t ) we see that an asymptotic formula for a follows if one is able to determine the ı θ X behaviour of E(e ) as t → R. (Recall that t → R,as n →∞, see Sect. 2.1.2.) Of course, once that has been achieved one still needs to determine the t and also appropriate expressions for the f (t ) and the σ(t ), which, in general, is not so direct. n n In any case, the plan above is actually the route to go. 3 Gaussian Khinchin Families In this section we discuss the notions of Gaussian and strongly Gaussian Khinchin families, obtain Hayman’s asymptotic formula for its coefﬁcients and, ﬁnally, intro- duce the class of Hayman functions, which amounts to a criterion for being strongly Gaussian. As we have seen, Theorem A, strongly Gaussian power series are Gaussian; and as we shall see, Theorem 3.8, power series in the Hayman class are strongly Gaussian. 3.1 Gaussian Khinchin Families Deﬁnition 3.1 (Gaussian power series) A power series f ∈ K and its Khinchin family ( X ) are termed Gaussian if X converges, as t ↑ R, in distribution to the t t ∈[0, R) t standard normal or, equivalently, if ı θ X −θ /2 lim E(e ) = e , for each θ ∈ R . t ↑ R For the exponential function f (z) = e , we deduce directly by taking the limit as t ↑ ∞ in formula (2.8) of the characteristic function of its normalized family, with θ ∈ R ﬁxed, that e is Gaussian. This fact means in particular that the normalized version of the Poisson random variable X , that is, ( X − t )/ t, converges in distribution to the t t standard normal. 123 864 A. Cantón et al. The function f (z) = 1/(1 − z) is not Gaussian. By taking limits in formula (2.7) as t ↑ 1, with θ ∈ R ﬁxed, we obtain that −ı θ ı θ X lim E(e ) = , for each θ ∈ R . t ↑1 1 − ı θ This actually means that X converges in distribution towards a variable Z, where Z + 1 is an exponential variable of parameter 1. The function f (z) = 1 + z is not Gaussian, either. By taking limits in formula (2.6) as t ↑∞, with θ ∈ R ﬁxed, we obtain that ı θ X lim E(e ) = 1 , for each θ ∈ R , t ↑∞ In other terms, X converges in distribution towards the constant 0. 3.1.1 A Criterion for Gaussianity The following simple criterion for gaussianity of functions f ∈ K non-vanishing in D(0, R), and in terms of the auxiliary function F of Sect. 2.1.5, is implicit in Hayman’s paper, see [16,Lem.4]. Theorem 3.2 (A criterion for gaussianity) If f ∈ K has radius of convergence R > 0 and vanishes nowhere in D(0, R), and if for the auxiliary function F one has sup | F (s + i φ)| φ∈R lim = 0 , (3.1) 3/2 s↑ln R F (s) then f is Gaussian. Proof. Since f vanishes nowhere in D(0, R), the auxiliary function F (z) = ln f (e ) is deﬁned and is holomorphic in the whole half-plane {z ∈ C : Rez < ln R}.Wehave, for s < ln R and θ ∈ R, that 2 3 θ |θ | F (s + ıθ) − F (s) − F (s)ı θ + F (s) ≤ sup F (s + i φ) · 2 6 φ∈R 2 s s Since F (s) = m(e ) and F (s) = σ (e ) for each s < ln R, writing e = t and substituting θ by θ/σ (t ) we deduce that sup F (s + i φ) m(t ) θ |θ | φ∈R ıθ/σ (t ) ln f (te ) − ln f (t ) − ı θ + ≤ , σ(t ) 2 σ (t ) 6 or, equivalently, for t = e with s < ln R and θ ∈ R, 123 Khinchin Families and Hayman Class 865 sup F (s + i φ) θ |θ | ˘ φ∈R ı θ X ln E(e ) + ≤ · (3.2) 3/2 2 F (s) 6 3.1.2 Some Applications of the Use of the Criterion of Theorem 3.2 Recall the auxiliary function F (z) = ln f (e ) from Sect. 2.1.5. For the exponential function f (z) = e , the auxiliary function is given by F (z) = e , and the gaussianity of f follows readily from Theorem 3.2. j R(z) Similarly, if R(z) = b z is a polynomial of degree N such that e ∈ K, j =0 R(z) z then e is Gaussian. For in this case F (z) = R(e ), and for z = s + ı φ we have that N N 3 jz 3 js Ns | F (z)|= b j e ≤ |b | j e = O(e ), j j j =0 j =0 while 2 js 2 Ns () F (s) = b j e ∼ b N e as s ↑∞ , j N j =0 and gaussianity follows from Theorem 3.2. Observe that () implies that b is real and b > 0; besides, since e ∈ K, the coefﬁcients of the polynomial R must be real numbers. For the partition function P(z) = 1/(1 − z ), the auxiliary function F may j =1 be written as kj z F (z) = e , for Rez < 0 . j ,k≥1 For z such that Rez < 0 and integer q ≥ 1, the qth derivative of F is given by (q) q−1 q kj z F (z) = k j e . j ,k≥1 For z = s + ı φ, with s < 0 and φ ∈ R, we then have that 2 3 kj s F (s + ı φ) ≤ k j e = F (s). j ,k≥1 Condition (3.1) requires one to check that F (−s) lim = 0 . 3/2 s↓0 F (−s) 123 866 A. Cantón et al. Let us see. For s > 0 and integer q ≥ 1, we have that q+1 (q) q−1 q −k( js) s F (−s) = k s ( js) e , k≥1 j ≥1 and so ∞ ∞ q+1 (q) q−1 q −kx q − y lim s F (−s) = k x e dx = y e dy s↓0 k 0 0 k≥1 k≥1 = ζ(2)(q + 1), and, thus, (q) F (−s) ∼ ζ(2)(q + 1) , as s ↓ 0 . (3.3) q+1 In particular, F (−s) 3 1/2 ∼ s , as s ↓ 0 . 3/2 F (−s) 2ζ(2) We conclude that the partition function P is Gaussian. Likewise, one readily veriﬁes that the ogfs Q and P are Gaussian. a,b For the inﬁnite product W (z) with a ≥ 1, b ≥ 0, let F (z) be the corresponding a,b auxiliary function. The following asymptotic formula for the qth derivative of F : a,b 1 b + 1 b + 1 1 (q) F (−s) ∼ ζ 1 + q + , as s ↓ 0 , (3.4) a,b q+(b+1)/a a a a s may be obtained with an argument very much like the one above for the partition function P. We deduce that F (−s) a,b (b+1)/(2a) s , as s ↓ 0 , 3/2 F (−s) a,b thus showing that each W , with a ≥ 1, b ≥ 0, is Gaussian. 3.1.3 A Criterion of Gaussianity for f = e in Terms of g Not unusually, we start with a power series g with non-negative coefﬁcients and are actually interested in its exponential f = e . The symbolic method of Combinatorics, see Flajolet–Sedgewick [9, Ch. II], gives that if g is the exponential generating function (egf) of a combinatorial class G of labelled objects with g(0) = 0, then e is the egf of the class of sets formed with the objects of G. For instance, e −1 • e is the egf of the class of sets of non-empty sets (or partitions of a set), see [9, p. 107], to be discussed in Sect. 5.2.1, 123 Khinchin Families and Hayman Class 867 ze • e is the egf of the class of sets of pointed sets (or idempotent functions), see [9, p. 131], to be discussed in Sect. 5.2.2, z/(1−z) • e is the egf of the class of sets of permutations (or ‘fragmented permuta- tions’), see [9, p. 125], to be discussed in Sect. 5.3. Observe that in the ﬁrst two instances R =∞, while in the last one, R = 1. The following theorem, a consequence of Theorem 3.2, exhibits conditions on g which imply that f = e is Gaussian. Theorem 3.3 (A criterion of gaussianity for f = e in terms of g) Let f ∈ K be such that f = e , where g has radius of convergence R > 0 and non-negative coefﬁcients. If g is a polynomial of degree 1 or g satisﬁes g (t ) lim = 0 , (3.5) 3/2 t ↑ R g (t ) then f is Gaussian. The three examples above are readily seen to be Gaussian as a consequence of Theorem 3.3. Proof We will assume that g is not a polynomial of degree 1 and that (3.5) is satisﬁed. z z The auxiliary function F is given by F (z) = ln f (e ) = g(e ),for Rez < ln R. Write g(z) as g(z) = b z , with b ≥ 0, for |z| < R. n n n=0 ( j ) j nz For an integer j ≥ 1, we have that F (z) = n b e ,for Rez < ln R.If n=1 s 3 9 we write t = e , with s < ln R, and let φ ∈ R, by using that n ≤ n(n − 1)(n − 2) for n ≥ 3, we have that ∞ ∞ 3 n 2 3 n−3 | F (s + ı φ)|≤ n b t ≤ b t + 8b t + t n(n − 1)(n − 2)b t n 1 2 n n=1 n=3 2 3 ≤ b t + 8b t + t g (t), 1 2 and by observing that n ≥ n(n − 1), also that F (s) ≥ t g (t). Therefore, 2 9 3 sup | F (s + ı φ)| b t + 8b t + t g (t ) 1 2 φ∈R () ≤ . 3/2 3 3/2 F (s) t g (t ) If R =∞, condition (3.1) of Theorem 3.2 follows from (), hypothesis (3.5) and the fact that lim g (t)> 0. t ↑ R If R < +∞, then () lim g (t ) =+∞. For otherwise, because of (3.5), we t ↑ R would have that lim g (t ) = 0, so that g ≡ 0 and g would be a polynomial of t ↑ R degree at most 2 and R =∞.From (), () and condition (3.5), we conclude that condition (3.1) of Theorem 3.2 is satisﬁed. 123 868 A. Cantón et al. 3.2 Strongly Gaussian Khinchin Families Following Báez-Duarte [2], we introduce the notion of strongly Gaussian power series. Deﬁnition 3.4 (Strongly Gaussian power series) A power series f ∈ K with radius of convergence R and its Khinchin family ( X ) are termed strongly Gaussian t t ∈[0, R) if ı θ X −θ /2 (a) lim σ(t ) =+∞ and (b) lim E(e ) − e dθ = 0 . t ↑ R t ↑ R |θ |<π σ (t ) The exponential function f (z) = e is strongly Gaussian. Recall that in this case σ(t ) = t. Strong gaussianity of e follows from its gaussianity and dominated convergence using the bound 2 2 ı θ X t (cos(θ / t )−1) −2θ /π E(e ) = e ≤ e , for θ ∈ R, t > 0 such that |θ | <π t . Theorem A below is both a local and a non-local central limit theorem satisﬁed by strongly Gaussian power series. It appears in Hayman’s [16] as Theorem I and Theorem II under the stronger hypothesis that f is in the Hayman class (to be discussed shortly in Sect. 3.3), but the proofs in [16] work for strongly Gaussian power series. Theorem A (Hayman’s central limit theorem) If f (z) = a z in K is strongly n=0 Gaussian, then a t 2 2 −(n−m(t )) /(2σ (t )) lim sup 2πσ (t ) − e = 0 . (3.6) t ↑ R f (t ) n∈Z Besides, lim P( X ≤ b) = (b), for every b ∈ R . t ↑ R And so, X converges in distribution towards the standard normal, and f is Gaussian. In this statement a = 0, for n < 0. By considering n =−1in (3.6), it follows that lim (m(t )/σ (t )) =+∞. In particular, this means that M given by (2.2)is t ↑ R f M =∞ for every f ∈ K that is strongly Gaussian. Theorem B (Hayman’s asymptotic formula) If f (z) = a z in K is strongly n=0 Gaussian, then 1 f (t ) a ∼ √ , as n →∞ , (3.7) t σ(t ) 2π n where t is given by m(t ) = n, for each n ≥ 1. n n 123 Khinchin Families and Hayman Class 869 This asymptotic formula is obtained by using Hayman’s identity (2.10) as follows. Write 2π a t σ(t ) 1 n n ˘ ı θ X − 1 = √ E(e ) dθ − 1 f (t ) n 2π |θ |<π σ (t ) 1 1 2 2 ı θ X −θ /2 −θ /2 = √ E(e ) − e dθ − √ e dθ, 2π 2π |θ |<π σ (t ) |θ |≥πσ (t ) n n and observe that conditions b) and a), respectively, of strong gaussianity (Deﬁni- tion 3.4), show that the ﬁrst and second terms of the expression above tend to 0 as n →∞. Actually, if ω is a good approximation of t , in the sense that n n m(ω ) − n lim = 0 , n→∞ σ(ω ) then 1 f (ω ) a ∼ √ , as n →∞ . (3.8) ω σ(ω ) 2π n Formula (3.8) (and also formula (3.7)) follows immediately from (3.6) of Theo- rem A. For the exponential f (z) = e one has m(t ) = t and σ(t ) = t,for t ≥ 0, and t = n for n ≥ 1. The asymptotic formula above gives 1 1 e ∼ √ √ , as n →∞ , n! n n 2π that is Stirling’s formula. Remark 3.5 The function f (z) = e is Gaussian, because of Theorem 3.3. Its variance 2 2 function σ (t ) = 4t tends towards ∞ as t ↑∞,but f is not strongly Gaussian, since its Taylor coefﬁcients of odd order are null and do not satisfy the asymptotic formula (3.7). 3.2.1 Báez-Duarte Substitution In general, precise expressions for the t are rare, since inverting m(t ) is usually compli- cated. But, fortunately, in practice, one can do with a certain asymptotic approximation due to Báez-Duarte [2]. This is the content of Theorem C below. Suppose that f ∈ K is strongly Gaussian. Assume that m (t ) is continuous and monotonically increasing to +∞ in [0, R), and that m (t ) is a good approximation of m(t ) in the sense that m(t ) − m (t ) lim = 0 . (3.9) t ↑ R σ(t ) Let τ be deﬁned by m (τ ) = n, for each n ≥ 1. n n 123 870 A. Cantón et al. Theorem C (Substitution) With the notations above, if f (z) = a z in K is n=0 strongly Gaussian and (3.9) is satisﬁed, then 1 f (τ ) a ∼ √ , as n →∞ . τ σ(τ ) 2π n Moreover, if σ(t ) is such that σ(t ) ∼ σ(t ) as t ↑ R, we may further write 1 f (τ ) a ∼ , as n →∞ . (3.10) τ σ(τ ) 2π n Theorem C follows readily from (3.8). In Sect. 6.1, we will exhibit appropriate approximations m and σ of the mean and 2 b variance functions m and σ of the ogfs of partitions P, Q, P and W , satisfying a,b in each case condition (3.9). 3.3 Hayman Class Strongly Gaussian functions in K have excellent asymptotic properties: the central limit Theorem A and the asymptotic formula (3.7) for its coefﬁcients. Thus far, f (z) = e is the only strongly Gaussian function encountered here. The class of Hayman consists of power series f in K which satisfy some concrete and veriﬁable conditions which imply that f is strongly Gaussian, see Theorem 3.8. Deﬁnition 3.6 (Hayman class) A power series f ∈ K is in the Hayman class if (variance condition) : lim σ(t ) =∞ , (3.11) t ↑ R and if for a certain function h:[0, R) → (0,π ], which we refer to as the cut (between a major arc and a minor arc), there hold ı θ X θ /2 (major arc) : lim sup E(e ) e − 1 = 0, (3.12) t ↑ R |θ |≤h(t)σ(t ) ı θ X (minor arc) : lim σ(t ) sup |E(e )|= 0 , (3.13) t ↑ R h(t )σ (t )<|θ |≤πσ (t ) The cut h of a f in the Hayman class is not uniquely determined. The functions in the Hayman class are called admissible by Hayman [16]; they are also called Hayman-admissible, or just H-admissible. Unlike other accounts (including Hayman’s), we do not allow a ﬁnite number of the coefﬁcients of the power series to be negative. For f in the Hayman class, the characteristic function of X is uniformly approxi- −θ /2 mated by e in the major arc, while it is uniformly o(1/σ (t )) in the minor arc. 123 Khinchin Families and Hayman Class 871 Observe that condition (3.13) may be written in terms of f or X as the requirement that ı θ | f (te )| ı θ X lim σ(t ) sup = 0 or lim σ(t ) sup E(e ) = 0 . t ↑ R f (t ) t ↑ R h(t )<|θ |≤π h(t )<|θ |≤π Lemma 3.7 If f ∈ K is in the Hayman class with cut h, then lim h(t)σ (t ) =∞ . t ↑ R ı θ X θ /2 t t Proof For θ = h(t)σ (t ), condition (3.12) implies that lim E(e ) e = 1; t t ↑ R ı θ X θ /2 t t while condition (3.13) implies that lim σ(t ) E(e ) = 0 . Thus, lim e /σ (t ) t ↑ R t ↑ R =∞ . As lim σ(t ) =∞, the conclusion follows. t ↑ R The conditions for a power series to be in the Hayman class amount to a criterion to be strongly Gaussian: Theorem 3.8 Power series in the Hayman class are strongly Gaussian. Proof Let us denote ı θ X −θ /2 I = E(e ) − e dθ, for t ∈ (0, R). |θ |<π σ (t ) Divide the integral I into an integral J over the major arc and an integral K over the t t t minor arc: ı θ X −θ /2 J = E(e ) − e dθ, |θ |≤h(t )σ (t ) ı θ X −θ /2 K = E(e ) − e dθ. h(t )σ (t )<|θ |≤πσ (t ) Bound K as ı θ X −θ /2 K ≤ 2πσ (t ) sup E(e ) + e dθ. h(t )σ (t )<|θ |≤πσ (t ) |θ |≥h(t )σ (t ) The ﬁrst summand of the bound of K tends to 0, as t ↑ R, because of the hypoth- esis (3.13), while the second summand tends to 0, as t ↑ R, since, as Lemma 3.7 dictates, lim h(t )σ (t ) =+∞. t ↑ R Bound J as 2 2 −θ /2 ı θ X θ /2 J = e E(e ) e − 1 dθ |θ |≤h(t )σ (t ) 2 2 −θ /2 ı θ X θ /2 ≤ e dθ sup E(e ) e − 1 R |θ |≤h(t )σ (t ) 123 872 A. Cantón et al. This bound of J tends to 0 as t ↑ R, by virtue of the hypothesis (3.12). As a consequence of Theorem 3.8, for asymptotic estimation of coefﬁcients of func- tions of the Hayman class, one may use the more ﬂexible asymptotic formula (3.10) of Báez-Duarte instead of the proper Hayman’s asymptotic formula (3.7). This fact will be crucial later, particularly for the arguments pertaining the partition functions of Sect. 6. 3.3.1 Combination and Perturbation of Functions of Hayman Class Maybe the reason for the relevance of the Hayman class in combinatorial questions is that it enjoys some closure and small perturbation properties. For instance, see, of course, Hayman [16]: (a) If f and g are in the Hayman class, then the product f · g is in the Hayman class, see [16, Thm. VII]. (b) If f is in the Hayman class, then e is in the Hayman class, see [16, Thm. VI]. (c) If f is in the Hayman class and if R is a polynomial in K, then the product R · f is in the Hayman class, see [16, Thm. VIII]. N n (d) Let B(z) = b z be a non-constant polynomial with real coefﬁcients such n=0 that for each d > 1 there exists m, not a multiple of d, such that b = 0, and such B B that if m(d) is the largest such m, then b > 0. If e is in K, then e is in the m(d) Hayman class, see [16,Thm.X]. 4 Exponentials and Hayman Class Let g be a non-constant power series with non-negative coefﬁcients and radius of convergence R > 0. The exponential f = e of such a g is in K. As mentioned above, see Sect. 3.1.3, this setting is quite usual in combinatorial questions. Write g(z) = b z , with b ≥ 0for n ≥ 0, and let f ∈ K be given by n n n=0 f = e . We will exhibit in Theorem 4.1 below conditions on the function g which ensure that f is in the Hayman class so that we can obtain an asymptotic formula for its coefﬁcients. Equipped with Theorem 4.1 below and the substitution Theorem C, we will discuss in Sects. 5.1, 5.2 and 5.3 a number of asymptotic formulae for coefﬁcients, mostly (but not only) of combinatorial interest, for functions f of the form f = e with non-constant g with non-negative coefﬁcients. Let ( X ) be the Khinchin family of f . The mean and variance functions of f , t t ∈[0, R) written in terms of g,are m(t ) = tg (t ) and σ (t ) = tg (t ) + t g (t), for t ∈ (0, R). Since g has non-negative coefﬁcients, the variance function σ (t ) of f is increasing in [0, R). The auxiliary function F of f is the function F (z) = g(e ),for z such that Rez < ln R. 123 Khinchin Families and Hayman Class 873 The variance condition (3.11) for belonging to the Hayman class in terms of g becomes lim tg (t ) + t g (t ) =+∞ . t ↑ R We want a cut function h for f . Let us start with the major arc. The bound (3.2) gives that sup F (s + ı φ) θ |θ | ˘ φ∈R ı θ X s ln E(e ) + ≤ , for t = e and s < ln R . 2 σ (t ) 6 The proof of Theorem 3.3 gives also that 2 3 | F (s + ı φ)|≤ b t + 8b t + t g (t), for t = e , s < ln R and φ ∈ R . 1 2 Deﬁne 2 3 ω (t ) = b t + 8b t + t g (t), for t ∈ (0, R). (4.1) g 1 2 z |z| Consider a potential cut function h:[0, R) → (0,π ]. Using that |e − 1|≤|z|e for all z ∈ C, we obtain that ı θ X θ /2 (t ) sup E(e ) e − 1 ≤ (t ) e , |θ |≤h(t)σ(t ) where (t ) is given by (t ) = ω (t ) h(t ) , for t ∈ (0, R). So, for h to fulﬁll condition (3.12) on the major arc, it is enough to have that lim ω (t ) h(t ) = 0 . t ↑ R In terms of h and g, condition (3.13) on the minor arc becomes ı θ lim σ(t ) exp sup Reg(te ) − g(t ) = 0 . t ↑ R h(t )≤|θ |≤π Thus, Theorem 4.1 Let g be a non-constant power series with radius of convergence R and non-negative coefﬁcients satisfying the variance condition (variance condition) : lim tg (t ) + t g (t ) =+∞ . (4.2) t ↑ R 123 874 A. Cantón et al. If there is a cut function h:[0, R) → (0,π ] satisfying (major arc) : lim ω (t ) h(t ) = 0 , (4.3) t ↑ R and ı θ (minor arc) : lim σ(t ) exp sup Reg(te ) − g(t ) = 0 , (4.4) t ↑ R h(t )<|θ |≤π then f = e is in the Hayman class. The function ω appearing in the major arc condition is given by the formula (4.1). Regarding the variance condition. For ﬁnite R, the variance condition (4.2) becomes lim g (t ) =∞. t ↑ R For R =∞, the variance condition (4.2) is always satisﬁed, since otherwise we would have that lim g (t ) = 0 and g would be constant. t →∞ Regarding the major arc condition. For ﬁnite R, the major arc condition (4.3) becomes lim g (t ) h(t ) = 0. t ↑ R For R =∞,if g(z) is not a polynomial, the condition on the major arc (4.3) reduces to lim t g (t ) h(t ) = 0 , t →∞ while if g is a polynomial of degree k ≥ 2, then (4.3) reduces to k 3 lim t h(t ) = 0 . t →∞ 4.1 Some Lemmas for Minor Arc Estimates In applications of Theorem 4.1, the variance and major arc condition for a cut for f = e are usually straightforward to verify. It is always the minor arc which requires the most work. We collect next a few elementary estimates for the functions 1/(1 − z) on |z|= t for t ∈ (0, 1) and integer k ≥ 1 which will prove useful in a number of veriﬁcations of the minor arc condition (4.4) of Theorem 4.1. Lemma 4.2 For t ∈ (0, 1) and θ ∈ R, −1 2 2 θ 4t sin 1 − t (1 − t ) = = 1 + . ı θ 2 2 1 − te (1 − t ) − 2t (cos θ − 1) (1 − t ) 123 Khinchin Families and Hayman Class 875 Lemma 4.3 Let ω ∈ (0,π) and t ∈ (0, 1). Then 1 − t 1 − t 1 − t 1 − t sup Re = Re and sup = . ı θ ı ω ı θ ı ω 1 − te 1 − te 1 − te 1 − te ω≤|θ |≤π ω≤|θ |≤π Proof The Möbius transformation T (z) = 1/(1 − z) carries the circle |z|= r, with r ∈ (0, 1), onto the circle orthogonal to the positive real axis through 1/(1 + r ) and ı φ 1/(1 − r ). For each r ∈ (0, 1), we have that Re((1 − r )/(1 − re )) decreases from 1to (1 − r )/(1 + r ) as φ increases from 0 to π (and as φ decreases from 0 to −π). Monotonicity of modulus follows from Lemma 4.2. Lemma 4.4 Let h(t ) be deﬁned in the interval (0, 1) with values in (0,π) and such that lim h(t )/(1 − t ) = 0. Let k be an integer k ≥ 1. Then we have t ↑1 (1 − t ) 1 − t k lim − 1 =− · ıh(t ) t ↑1 h(t ) 1 − te 2 Proof. Lemma 4.2 and lim h(t ) = 0 give that t ↑1 1 − t () lim = 1 . ıh(t ) t ↑1 1 − te Lemma 4.2 gives that h(t ) 4t sin 1 − t 2 () 1 − ∼ , as t → 1 . ıh(t ) 2 1 − te (1 − t ) The case k = 2 follows from (). The general case follows from combining (),the case k = 2 and that y − 1 k lim = · y→1 y − 1 2 5 Some Examples of Exponentials in the Hayman Class In this section we verify that relevant examples of f = e are in the Hayman class. Some of them have been mentioned already, see the discussion of Sect. 3.1.3, like z z g(z) = e − 1 (for non-empty sets) or g(z) = ze (for pointed sets). We shall consider too the example of a polynomial g with non-negative coefﬁcients or the case g(z) = (1 + z)/(1 − z). The function g in all these cases has a closed formula which facilitates the veriﬁ- cation of the conditions of Theorem 4.1, and thus the corroboration that f is in the 123 876 A. Cantón et al. Hayman class and that Hayman’s asymptotic formula (3.7) applies. Besides, the mean function m(t ) and the variance function σ(t ) have as well, in all these cases, closed formulas, and the asymptotic behaviour of t and f (t ) and σ(t ) appearing is (3.7) n n is quite direct. 5.1 Exponential of a Polynomial Let B(z) = b z be a polynomial of degree N ≥ 1 with non-negative coef- n=0 ﬁcients. Assume further that gcd{1 ≤ n ≤ N : b > 0}= 1. We will check that f = e is in the Hayman class. For this aim, we could appeal to the general result d) of Hayman of Sect. 3.3.1. But the present case of non-negative coefﬁcients may be neatly cast as a consequence of Theorem 4.1, and we include a proof. N j Proposition 5.1 Let B(z) = b z be a polynomial of degree N ≥ 1 with non- j =0 negative coefﬁcients and such that q gcd{1 ≤ n ≤ N : b > 0}= 1 . (5.1) B(z) Then, f (z) = e is in the Hayman class. Condition (5.1) is indispensable. In fact, if q > 1, then f = e is not strongly Gaussian, since then f (z) = H (z ), for a certain entire power series H, and the coef- ﬁcients of f can not satisfy Hayman’s asymptotic formula (3.7) as strongly Gaussian functions do. Proof As mentioned above, since the radius of convergence of B(z) is R =∞,the variance condition (4.2) is satisﬁed. −α We verify next that h(t ) given by h(t ) = min{π/ N , t }, with α ∈ ( N /3, N /2), satisﬁes conditions (4.3) and (4.4) of Theorem 4.1 for a cut. Fix α in that interval. Condition (4.3) on the major arc reduces, as we have mentioned above, to N 3 lim t h(t ) = 0 , t →∞ which is satisﬁed since α> N /3. Now, the minor arc. Notice that ı θ n Re B(te ) − B(t ) = b t (cos(nθ) − 1). n=1 All the summands above are non-positive. We distinguish now between θ’s close to and away from the N th roots of unity. ı θ Denote T ={e : θ ∈ R}= ∂ D. For t > 0 and 1 ≤ j < N , deﬁne the arcs in T 2π j ı θ I (t ) = e ∈ T : θ − < h(t ) . 123 Khinchin Families and Hayman Class 877 and for t > 0 deﬁne also ı θ I (t ) = e ∈ T :|θ | < h(t ) . These I (t ),0 ≤ j < N are arcs in T of length 2h(t ) around the N th roots of unity. I is the major arc. Since h(t ) ≤ π/ N , the arcs I (t ) are disjoint. 0 j ı θ ˆ ˆ Denote also by I (t ) the arc I (t ) = e ∈ T :|θ | < Nh(t ) . 0 0 ı θ Let N ={1 ≤ n ≤ N : b > 0}. We will use that, since gcd(N ) = 1, no e with ı θ θ ∈ R, except e = 1, is simultaneously a nth root of unity for all n ∈ N . ı2π j / N For each 1 ≤ j < N , there is n ∈ N so that e , the center of the arc I (t ),is j j not a n th root of unity. Since lim h(t ) = 0, there is t > 1 and, for each 1 ≤ j < N , j t ↑∞ 0 a δ > 0, so that ı θ b (cos(n θ) − 1) ≤−δ , for e ∈ I (t ) and t ≥ t . n j j j 0 Let = min{δ , 1 ≤ j < N } and M = min N . Then n M ı θ b t (cos(n θ) − 1) ≤−t , for e ∈ I (t ) and 1 ≤ j < N and t ≥ t . n j j 0 N −1 ı θ ıN θ For e ∈ T \ I (t ) ∪ I (t ) we have that e ∈ T \ I (t ) and, thus, using j 0 0 j =1 2 2 the inequality 1 − cos x ≥ 2x /π , valid for x ∈[−π, π ], that 2 2 cos( N θ) − 1 ≤− N h(t ) , and so that N N 2 b t (cos( N θ) − 1) ≤− t h(t ) , 2 2 with = (2/π ) N b . Therefore, if h(t ) ≤|θ|≤ π and t > t , we have that ı θ n M N 2 Re B(te ) − B(t ) = b t (cos(nθ) − 1) ≤ max −t , −t h(t ) . n=1 Since σ(t ) grows just as a power of t, condition (4.4) on the minor arc is satisﬁed since α< N /2. We conclude that e is in the Hayman class. 5.1.1 Frequency of Permutations of Given Order As a ﬁrst example of the efﬁcient alliance of strong gaussianity with the substitution Theorem C, we now give, for any ﬁxed integer k ≥ 1 and as n →∞, an asymptotic formula for the number A (n) of permutations σ ∈ S such that σ is the identity. k n Observe, for instance, that A (n) is the number of involutions in S . The particular 2 n 123 878 A. Cantón et al. cases k = 2 and k prime of this asymptotic formula are due to Moser–Wyman, [23,24]. For general k, the formula is essentially due to Wilf [28]. The permutations counted by A (n) are those σ ∈ S so that all the cycles in its k n cycle decomposition have lengths which divide k. Thus, the symbolic method, see [9, p. 124], gives us that the egf I (z) of these permutations is ⎛ ⎞ A (n) 1 n d ⎝ ⎠ I (z) = z = exp z . n! d n=0 1≤d|k Denote R (z) = z . 1≤d|k As I (z) is the exponential of the polynomial R (z) and gcd{d ≥ 1 : d | k}= 1, we k k have by Proposition 5.1 that I (z) is in the Hayman class and, in particular, that I (z) k k is strongly Gaussian. By applying the asymptotic formula (3.10) of the substitution Theorem C, we will obtain the asymptotic formulae (5.2) and (5.3)for A (n), distinguishing between k odd or k even. Denote by m (t ) and σ (t ),for t > 0, the mean and variance functions of the Khinchin family of I (t ). Observe that d 2 d m (t ) = t and σ (t ) = dt , for t > 0 . 1≤d|k 1≤d|k k/2 We take σ (t ) = kt ,for t > 0. Observe that σ (t ) ∼ σ (t ),as t →∞. k k k To properly deﬁne an approximation m (t ) of m (t ), we distinguish between k even k k and k odd. (a) Case k odd. The divisors d of k, different from k, satisfy d ≤ k/3. Thus m (t ) = k k/3 k t + O(t ),as t →∞, and if we deﬁne m (t ) = t , then k/3 m (t ) − m (t ) = O(t ) = o( σ (t )) , as t ↑∞ . k k k 1/k For n ≥ 1, we take τ = n , so that m (τ ) = n. Strong gaussianity and the n k n substitution Theorem C give for k ≥ 1, odd and ﬁxed, that 1/k R (n ) A (n) 1 1 e ∼ √ √ √ , as n →∞ . (5.2) n/k n! n n 2π k (b) Case k even. Now a divisor of k is k/2, which is the order of σ (t ).Wehave k k/2 k/3 m (t ) = t + t + O(t ),as t ↑∞, and we take 1 1 k k/2 k/2 m (t ) = t + t + = t + , for t > 0 . 4 2 123 Khinchin Families and Hayman Class 879 Thus, k/3 m (t ) − m (t ) = O(t ) = o( σ (t )) , as t ↑∞ . k k k 2/k Deﬁne τ = ( n − 1/2) , so that m (τ ) = n, for each n ≥ 1. Since n k n √ √ 1 1 n ln n − = n ln n + n ln 1 − √ 2 2 n n 1 1 1 = ln n − n √ + + O 3/2 2 2 n 8n n n n 1 = ln n − − + o(1), as n →∞ , 2 2 8 and so n/2 − n/2 −1/8 n − ∼ n e e , as n →∞ , we have that n n/k − n/k −1/(4k) τ ∼ n e e , as n →∞ . Besides, √ √ k/2 σ (τ ) = k τ ∼ k n , as n →∞ . k n n Strong gaussianity and substitution give for k ≥ 1, even and ﬁxed, that 1/k 1/(4k) R ((n− n+1/4) ) A (n) 1 e e n/k ∼ √ √ e √ , as n →∞ . (5.3) n/k n! n n 2π k Using (5.2) and (5.3), ones sees that if q > k, then A (n) 1 1 ln = − n ln n + O(n), as n →∞ , A (n) q k and, thus, irrespective of the parity of q and k, A (n) () lim = 0 , n→∞ A (n) Besides, lim A (n)/n!= 0 and lim A (n) =∞. n→∞ k n→∞ k For the number B (n) of permutations of (1,..., n) whose order is exactly k, one obtains via Möbius inversion and (), see Wilf [28], that B (n) ∼ A (n), as n →∞ . k k 123 880 A. Cantón et al. 5.2 Entire Functions: Bell Numbers and Idempotent Functions We apply next Theorems 4.1 and C to the egfs of the Bell numbers and of the number of idempotent functions, which are both entire functions. 5.2.1 Bell Numbers e −1 Consider ﬁrst the function f (z) = e , the egf of the Bell numbers B described above in Sect. 2.1.6. Result b) of Hayman in Sect. 3.3.1 already gives that f is in the Hayman class, but let us see now how this fact follows most easily from Theorem 4.1 on our way to obtain the Moser–Wyman asymptotic formula (5.5) below for the Bell numbers, [22]. e −1 t 2 t For f (z) = e , we have that m(t ) = te and that σ (t ) = t (t + 1)e .Inthe notation of Theorem 4.1, the function g is g(z) = e − 1. For a potential cut h, condition (4.3) for the major arc requires that 3 t 3 lim t e h(t ) = 0 . t ↑∞ For the minor arc, using that t cos θ t t 2 e − e ≤− e θ , for any t ≥ 1 and |θ | <π , (5.4) 2π we obtain, for t ≥ 1, that ı θ ı θ te t t cos θ t t 2 Reg(te ) − g(t ) ≤|e |− e = e − e ≤− e θ . 2π And so ı θ t 2 exp sup Reg(te ) − g(t ) ≤ exp − e h(t ) . 2π h(t )≤|θ |≤π −αt With h(t ) = e ,for t > 0 and ﬁxed α ∈ (1/3, 1/2), both conditions (4.3) e −1 and (4.4) are satisﬁed, and we conclude that e is in the Hayman class. The t for f are such that m(t ) = t e = n. Thus t = W (n), where W is the n n n n Lambert function. Since f is strongly Gaussian, formula (3.7) gives that W (n) e −1 B 1 e ∼ √ , as n →∞ , (5.5) W (n) n n! 2π W (n)(W (n) + 1)e W (n) which is the asymptotic formula for the Bell numbers of Moser–Wyman, [22]. 123 Khinchin Families and Hayman Class 881 5.2.2 Counting Idempotent Functions Consider now the function ze n f (z) = e = z , n! n=0 which turns out to be the egf of idempotent functions. That is, U denotes the number of idempotent functions u of {1,..., n} into itself, i.e., functions u such that u ◦ u ≡ u. See [9, p. 571]. We will verify shortly that f is in the Hayman class, towards proving the Harris– Schoenfeld ([14,15]) asymptotic formula (5.6) below for the U . For that aim we could appeal to the combination of results b) and c) of Hayman of Sect. 3.3.1, but we favour a direct proof via Theorem 4.1. z 2 t 2 3 2 t For f (z) = exp(ze ), we have that m(t ) = (t + t )e and σ (t ) = (t + 3t + t )e , for t > 0. The function g is, in this case, g(z) = ze . For a potential cut h towards applying Theorem 4.1, observe that condition (4.3) for the major arc requires that 4 t 3 lim t e h(t ) = 0 . t ↑∞ For the minor arc, using again (5.4), we have, for t > 1, that ı θ ı θ t cos θ ıt sin θ t t cos θ t t 2 Reg(te ) − g(t ) ≤|te e e |− te = te − te ≤− te θ , 2π so that ı θ t 2 exp sup Reg(te ) − g(t ) ≤ exp − te h(t ) , for t > 1 . 2π h(t )≤|θ |≤π −αt As before, with α ∈ (1/3, 1/2) and h(t ) = e ,for t > 0, both (4.3) and (4.4) are fulﬁlled, and we deduce from Theorem 4.1 that f is strongly Gaussian. We now obtain the promised asymptotic formula for the U . Since f is strongly Gaussian, we obtain, with t given by m(t ) = t (t + 1)e = n, for each n ≥ 1, that n n n n U 1 1 n ∼ √ exp , as n →∞ , (5.6) n+3/2 n! t /2 1 + t 2π n e t n the asymptotic formula due to Harris–Schoenfeld [14,15]. 123 882 A. Cantón et al. 5.3 Functions in D:Cover Mapof {z ∈ C :|z| > 1} Consider now 1 + z f (z) = exp = A z , for z ∈ D . 1 − z n=0 This function f is a universal cover of {z ∈ C :|z| > 1}. In this case R = 1, and the function g is given by g(z) = (1 + z)/(1 − z),for z ∈ D. The mean and variance functions of f are given by 2t 2t (1 + t ) m(t ) = and σ (t ) = for t ∈ (0, 1). 2 3 (1 − t ) (1 − t ) The variance condition (4.2) is clearly satisﬁed. Now we look for a cut function h of the form h = (1 − t ) with some appropriate α> 0, towards applying Theorem 4.1. The condition on the major arc (4.3) reduces 3 4 to lim h(t ) /(1 − t ) = 0. For the exponent α, this translates into requiring that t ↑1 α> 4/3. For the minor arc, we start by writing g as g(z) = − 1 . 1 − z If the exponent α in the deﬁnition of h satisﬁes α> 1, then lim h(t )/(1 − t ) = 0, t ↑1 and then Lemmas 4.3 and 4.4 give that (1 − t ) ı θ lim sup sup Re g(te ) − g(t ) h(t ) t ↑1 h(t )≤|θ |≤π (1 − t ) 1 1 ≤ 2 lim − =−1 . 2 ıh(t ) t ↑1 h(t ) 1 − te 1 − t And so, for some t ∈ (0, 1) and δ> 0, we have that h(t ) ı θ sup Re(g(te ) − g(t )) ≤−δ , for t ∈ (t , 1). (1 − t ) h(t )≤|θ |≤π 3/2 Since σ(t ) 1/(1 − t ) ,as t ↑ 1, we deduce that (4.4) is satisﬁed as long as α< 3/2. Thus, with α ∈ (4/3, 3/2) for the cut function, we conclude that f is in the Hayman class. Next, by appealing to the substitution Theorem C, we obtain an asymptotic formula for the Taylor coefﬁcients A of f (z). 123 Khinchin Families and Hayman Class 883 2 3/2 Deﬁne m (t ) = 2/(1 − t ) and σ(t ) = 2/(1 − t ) ,for t ∈ (0, 1), and observe that σ(t ) ∼ σ(t), as t ↑ 1 , and that m (t ) − m(t ) = = o(σ (t )) , as t ↑ 1 , 1 − t and, thus, that condition (3.9) is satisﬁed. For each n ≥ 1, take τ given by 1 − τ = 2/n. Since n n −n τ = exp n ln √ = exp 2n + 1 + o(1) , as n →∞ , 1 − 2/n by virtue of formula (3.10) we deduce that 2 2 n 1 1 e A ∼ √ , as n →∞ . 1/4 3/4 2 n 2π This asymptotic formula is Theorem XIII of Hayman’s [16]. See also Wright [30] and Macintyre–Wilson [19]. For the closely related egf z I exp = z 1 − z n! n=0 of the number I of fragmented permutations of (1,..., n) we have, with a similar z/(1−z) argument, that e is in the Hayman class and that I 1 1 2 n ∼ √ e , as n →∞ . 3/4 n! n 2 e π See [9, p. 563]. Remark 5.2 For the functions 1 1 f (z) = exp γ , for z ∈ D , (1 − z) 1 − z with integers β ≥ 0 and γ ≥ 1, considered by Wright [30] and Macintyre–Wilson in [19] (both with more general β and γ ), Theorem 4.1 gives us that f is strongly Gaussian. The function g(z) is in this case 1 1 g(z) = β ln + γ · 1 − z 1 − z 123 884 A. Cantón et al. For the estimate of the minor arc, just observe that 1 1 ı θ Reg(te ) − g(t ) ≤ γ Re − . ı θ 1 − te 1 − t Appropriate approximations m and σ are γ 2γ m (t ) = and σ (t ) = , 2 3 (1 − t ) (1 − t ) which satisfy condition (3.9). For the coefﬁcients a of f (z) we have 1/4−β/2 γ/2 1 γ e 1 2 γ n a ∼ √ √ e , as n →∞ . 3/4−β/2 2π 2 The functions f (z) = exp(1/(1 − z) ), for integer ρ> 0, also considered by Wright [32] (with more general exponents ρ), are also strongly Gaussian. A cut function h(t ) = (1 − t ) ,for α> 0, satisﬁes the condition on the major arc if 3α> ρ + 3. Using Lemmas 4.3 and 4.4 with k = ρ, we see that h satisﬁes the condition on the minor arc if 2α< 2 + ρ. ρ+1 1/(1+ρ) given by ρt /(1−t ) = n (so that 1−t ∼ (ρ/n) ,as n →∞), we With t n n n n have an asymptotic formula for the coefﬁcients of f , but unless ρ = 1, the substitution Theorem C is of no avail for simplifying the resulting expression. 6 Partitions Next we discuss ogfs f of different sorts of partitions: P for usual partitions, Q for partitions with distinct parts, P for partitions with parts in an arithmetic sequence, a,b 1 b or W , which codiﬁes plane partitions, or W for colored partitions. 1 1 As in Sect. 5, we will use Theorem 4.1 to verify that all these partition functions are strongly Gaussian and then use (3.10) to provide asymptotic formulas for their coefﬁcients. But in sharp contrast to the power series dealt with in Sect. 5, the corresponding functions g do not have closed formulas and the veriﬁcations of the conditions of Theorem 4.1 are more involved. In addition, the mean and variance functions do not have closed formulas either and this requires obtaining suitable approximations of these functions to ﬁnd an approximation τ of t . And, ﬁnally, a proper asymptotic n n formula for the a requires, in this setting, determining the asymptotic behaviour of f on (0, R) in order to be able to get an asymptotic formula for f (t ). The variance condition and the major arc requirement of Theorem 4.1 are quite direct; it is the minor arc condition which requires more work. We will obtain the needed approximations of means and variances in Sect. 6.1, the asymptotic expressions on (0, 1) in Sect. 6.2 and, ﬁnally, we will discuss strong gaussianity and asymptotic formulae of their coefﬁcients in Sect. 6.3. 123 Khinchin Families and Hayman Class 885 These asymptotic formulae of coefﬁcients a of partition functions turn out to be always of the form 1 δ γ n a ∼ α e , as n →∞ , for appropriate α, β, γ , δ > 0. 6.1 Approximation of Means and Variances We collect here convenient approximations of the mean and variance functions, satis- fying in each case condition (3.9) of Theorem C, of the Khinchin families of a number of partition functions. Consider ﬁrst the partition function P(z) = , for |z| < 1 , 1 − z j =1 with Khinchin family ( X ) and mean and variance functions t t ∈[0,1) ∞ ∞ j 2 j jt j t m(t ) = and σ (t ) = , for t ∈ (0, 1). j j 2 1 − t (1 − t ) j =1 j =1 The precise values of the integrals in the following lemma will be invoked frequently in the rest of the paper. Lemma 6.1 a) s ln ds = ζ(u + 2)(u + 1), for u > −1 , −s 1 − e ∞ −s b) s ds = ζ(u + 1)(u + 1), for u > 0 , −s 1 − e ∞ −s c) s ds = ζ(u)(u + 1), for u > 1 . −s 2 (1 − e ) −s It is convenient to consider m(e ),for s > 0, and then s ↓ 0. We have − js 1 e −s m(e ) = js · − js s (1 − e ) j =1 Let φ be the function −x xe φ(x ) = , for x > 0 . −x 1 − e 123 886 A. Cantón et al. If we set φ(0) = 1, then φ ∈ C [0, +∞) and φ(s)ds = ζ(2),byb)ofLemma 6.1. In addition, we have that lim φ(t ) = 0 and |φ (s)|ds < +∞.Fromthe Euler t →∞ summation formula (1.1) applied to φ, we get that ζ(2) −s sm(e ) − = O(1), as s ↓ 0 . (6.1) 2 −s and, therefore, that lim s m(e ) = ζ(2), and so s↓0 ζ(2) −s m(e ) ∼ , as s ↓ 0 . (6.2) −s 2 If we deﬁne m (e ) = ζ(2)/s ,for s > 0, we have that m(t ) ∼ m (t ),as t ↑ 1. For the variance function σ (t ) we obtain analogously, using Lemma 6.1, that (3)ζ (2) 2 −s σ (e ) ∼ , as s ↓ 0 . (6.3) 2 −s 2 3 Moreover, if we deﬁne σ (e ) = π /(3s ),for s > 0, we have that σ(t ) ∼ σ(t), as t ↑ 1 , and also, using (6.1), that −s −s m(e ) − m (e ) = O( s) = o(1), as s ↓ 0 . −s σ(e ) Next, we turn to Q, the ogf of partitions with distinct parts. For the mean and variance functions of its Khinchin family, using Lemma 6.1 again, plus the identity ln(1 + x ) = ln(1 − x ) − ln(1 − x ) for x ∈ (0, 1), and its ﬁrst two derivatives, we obtain that, as s ↓ 0, − js e ζ(2) −s −s m(e ) = j ∼ m (e ), − js 2 1 + e 2s j =1 (6.4) − js e ζ(2) 2 −s 2 2 −s σ (e ) = j ∼ σ (e ); − js 2 3 (1 + e ) s j =1 and, also, that −s −s m(e ) − m (e ) √ = O( s) = o(1), as s ↓ 0 . −s σ(e ) 123 Khinchin Families and Hayman Class 887 For the ogf of P with parts in the arithmetic progression {aj + b; j ≥ 0}, with a,b integers a, b ≥ 1, we have analogously that, as s ↓ 0, −(aj +b)s e ζ(2) −s −s m(e ) = (aj + b) ∼ m (e ), −(aj +b)s 2 1 − e as j =0 (6.5) −(aj +b)s e 2ζ(2) 2 −s 2 2 −s σ (e ) = (aj + b) ∼ σ (e ); −(aj +b)s 2 3 (1 − e ) as j =0 and, also, −s −s m(e ) − m (e ) √ = O( s) = o(1), as s ↓ 0 . −s σ(e ) For integers a ≥ 1 and b ≥ 0, consider the inﬁnite product W (z) = , for z ∈ D . 1 − z j =1 Let m(t ) and σ(t ) be the mean and variance functions of the Khinchin family of W . Consider the real function a −x x e φ(x ) = x , for x ≥ 0 , −x 1 − e whose derivatives of order less than b vanish at x = 0 and x =+∞, and is such that 1 b + 1 b + 1 φ(x ) dx = ζ 1 + 1 + . a a a Arguing as above for the case of regular partitions, using φ and Euler’s summation of order b + 1, we get that −s −s m(e ) = m (e ) + O , as s ↓ 0 , where 1 b + 1 b + 1 1 −s m (e ) = ζ 1 + 1 + · (6.6) 1+(b+1)/a a a a s Analogously, 1 b + 1 b + 1 1 2 −s 2 −s σ (e ) ∼ ζ 1 + 2 + σ (e ), as s ↓ 0 . 2+(b+1)/a a a a s (6.7) 123 888 A. Cantón et al. Observe that −s −s m(e ) − m (e ) (b+1)/2a = O(s ) = o(1), as s ↓ 0 . (6.8) −s σ(e ) 6.2 Asymptotics of Partition Functions on (0, 1) We start with the ogf of partitions P; an instance we will built upon. For the partition function P we have the well-known asymptotic formula in the interval (0, 1): ζ(2) 1 −s ln P(e ) = + ln s − ln 2π + o(1), as s ↓ 0 , s 2 and, thus, −s ζ(2)/s P(e ) ∼ √ se , as s ↓ 0 . (6.9) 2π See Hardy–Ramanujan [13, Sect. 3.2] (the formula for g(x ) there should have a factor 3/2 (1 − x ) instead of 1 − x). See also de Bruijn [8, Ch. 3, Ex. 3] or even Báez-Duarte [2, (2.21)]. We provide a proof since its ingredients are to be used below to handle other partition functions. Fix s > 0 and apply Euler’s summation of order 2 to the function f (x ) =− ln(1 − −sx e ). Observe that lim f (x ) = lim f (x ) = 0. Write x →∞ x →∞ ∞ ∞ 1 1 B ({x }) −s ln P(e ) = f ( j ) = f (x )dx + f (1) − f (1) − f (x ) dx . 2 12 2! 1 1 j =1 where B ( y) is the second Bernoulli polynomial. Now, using a) in Lemma 6.1 we have that ∞ ∞ 1 1 1 ln dx = ln dx −sx −x 1 − e s 1 − e 1 s ∞ s 1 1 1 1 = ln dx − ln dx −x −x s 1 − e s 1 − e 0 0 ζ(2) = + ln s − 1 + O(s), as s ↓ 0 . Also, 1 1 1 1 f (1) − f (1) =− ln s + + O(s), as s ↓ 0, 2 12 2 12 and ∞ ∞ 2 sx B ({x }) (sx ) e B ({x }) 1 2 2 f (x ) dx = dx . sx 2 2 2! (e − 1) 2! x 1 1 123 Khinchin Families and Hayman Class 889 2 y y 2 Since the function y → ( y e )/(e − 1) is bounded in [0, +∞) and tends to 1 as y ↓ 0, dominated convergence gives that ∞ ∞ B ({x }) B ({x }) 1 2 2 f (x ) dx = dx + o(1), as s ↓ 0 . 2! 2! x 1 1 Thus, ζ(2) 1 1 B ({x }) 1 −s ln P(e ) = + ln s − 1 + − dx + o(1), as s ↓ 0 . s 2 12 2! x We may identify the constant term of the above expression by using analogous Euler’s summation of order 2 between 1 and N for the function ln x, to obtain Stirling’s approximation in the following precise (and standard) form 1 1 1 B ({x }) 1 ln N!= N ln N − N + 1 + ln N + − 1 + dx , 2 12 N 2! x and thus deduce that 1 B ({x }) 1 1 − + dx = ln 2π. 12 2! x The same argument gives for the ogf Q of partitions into distinct parts that 1 ζ(2) −s Q(e ) ∼ √ exp , as s ↓ 0 . (6.10) 2s and for the ogf P of partitions with parts in the arithmetic progression {aj +b; j ≥ 0} a,b with integers a, b ≥ 1, that 1 b ζ(2) −s b/a−1/2 P (e ) ∼ √ (as) exp , as s ↓ 0 . (6.11) a,b a as 2π Recall that P ≡ P and P ≡ Q. The asymptotic formula for Q may be obtained 1,1 2,1 from the identity (2.9) and the corresponding asymptotic formula (6.9)for P. Now we turn to the ogf W , with integers a ≥ 1 and b ≥ 0. Fix s > 0. We will apply Euler’s summation of order b + 2 to the function f (x ) = x ln , for x > 0 . −sx 1 − e Note that b −s ln W (e ) = f ( j). j =1 123 890 A. Cantón et al. Observe that ∞ ∞ 1 1 1 dy (b+1)/a f (x )dx = y ln − y (b+1)/a a s 1 − e y 1 s 1 1 1 dy (b+1)/a = y ln (b+1)/a − y a s 1 − e y 1 1 1 dy (b+1)/a − y ln (b+1)/a − y a s 1 − e y 1 b + 1 b + 1 1 = ζ 1 + (b+1)/a a a a s 1 1 a − ln − + O(s). b + 1 s (b + 1) ( j ) For the function f at ∞, we have that lim f (x ) = 0, for j ≥ 0. x →∞ We need the values of f and its derivatives up to order b + 1at x = 1. We have 1 1 f (1) = ln = ln + O(s). −s 1 − e s Write 1/a f (x ) = g(s x ), b/a where 1 y b b+a b b g( y) = y ln = y + y ln − a ( y ln y) . a a − y y ! "# $ 1 − e e − 1 ! "# $ =η( y) =ω( y) Observe that ( j ) ( j ) 1/a f (1) = g (s ), for j ≥ 1. (b− j )/a The function ω is holomorphic near 0, and its Taylor expansion starts with b+a (1/2) y , while ⎛ ⎞ b! 1 ( j ) b− j ⎝ ⎠ η ( y) = y ln y + , for 1 ≤ j ≤ b and y > 0 . (b − j )! b + 1 − i i =1 (b+1) (b+2) 2 Also, η ( y) = b!/ y and η ( y) =−b!/ y ,for y > 0. And, also, (b+1) 1/a (b+1) 1/a 1/a (b+1) 1/a 1/a (b+1) 1/a f (1) = s g (s ) = s ω (s ) − as η (s ) =−b! a + O(s). 123 Khinchin Families and Hayman Class 891 With all this in mind, we have that b+1 1 B j +1 j +1 ( j ) ( j ) ( f (1) + f (∞)) + (−1) ( f (∞) − f (1)) 2 ( j + 1)! j =1 ⎛ ⎞ 1 1 B b! j +1 j +1 ⎝ ⎠ = ln + (−1) ln s 2 s ( j + 1)! (b − j )! j =1 b j B b! 1 B j +1 b+2 j +1 b+2 + a (−1) +a(−1) b!+ O(s). ( j + 1)! (b − j )! b + 1 − i (b + 2)! j =1 i =1 Applying Euler’s summation of order b + 1to x , with b ≥ 1, just in the inter- val [0, 1], and taking into account that its bth derivative is a constant (b!, actually), we deduce that b−1 1 1 b! j +1 j +1 1 = + + (−1) . b + 1 2 ( j + 1)! (b − j )! j =1 Thus, B b! B 1 1 j +1 b+1 j +1 b+1 (−1) = (−1) + − , ( j + 1)! (b − j )! b + 1 2 b + 1 j =1 for b ≥ 1, and for b = 0also. So, we may simplify and write b+1 1 B j +1 j +1 ( j ) ( j ) ( f (1) + f (∞)) + (−1) ( f (∞) − f (1)) 2 ( j + 1)! j =1 1 B b+1 b+1 = − + (−1) ln s b + 1 b + 1 b j B b! 1 B j +1 b+2 j +1 b+2 + a (−1) + a(−1) b!+ O(s). ( j + 1)! (b − j )! b + 1 − i (b + 2)! j =1 i =1 Finally, B ({x }) b+2 b+3 (b+2) (−1) f (x ) dx (b+2)! B ({x }) dx b+2 b+3 1/a 2 (b+2) 1/a = (−1) (s x ) g (s x ) . (b+2)! x 123 892 A. Cantón et al. 2 (b+2) The function y → y g ( y) is bounded in (0, ∞) and as y ↓ 0 tends to b! a.We deduce from dominated convergence that, as s ↓ 0, ∞ ∞ B ({x }) B ({x }) dx b+2 b+2 b+3 (b+2) b+3 (−1) f (x ) dx = (−1) b! a + o(1). (b + 2)! (b + 2)! x 1 1 From this, we have 1 b + 1 b + 1 1 B b+1 b −s b+1 ln W (e ) = ζ 1 + + (−1) ln s (b+1)/a a a a s b + 1 b j a B b! 1 j +1 b+1 − + a (−1) (b + 1) ( j + 1)! (b − j )! b + 1 − i j =1 i =1 B B ({x }) dx b+2 b+2 b+2 b+3 + a(−1) b!+ (−1) b! a + o(1). (b + 2)! (b + 2)! x The constant term of the above expression may be written more compactly by appealing to the so called (generalized) Glaisher–Kinkelin constants appearing in (in fact, deﬁned by) the asymptotic formula of (generalized) hyperfactorials. If we apply Euler’s summation of order b + 2 to the function x ln x between 1 and N , we obtain that 1 1 1 b b+1 b+1 b n ln n = N ln N − N + N ln N b + 1 (b + 1) 2 n=1 B b! j +1 j +1 b− j + (−1) N ln N ( j + 1)! (b − j )! j =1 b−1 B b! 1 j +1 j +1 b− j (‡) + (−1) N ( j + 1)! (b − j )! b − i + 1 j =1 i =1 B 1 B b! 1 b+1 j +1 b+1 j +1 + (−1) H + − (−1) b + 1 (b + 1) ( j + 1)! (b − j )! b − i + 1 j =1 i =1 B b! dx 1 b+2 b+2 b+3 − (−1) b!− (−1) B ({x }) + O . b+2 (b + 2)! (b + 2)! x N Here, H denotes the bth harmonic number. This expression (‡) is the analogue of Stirling’s formula but now for the hyper- factorials j of order b; the factorials being the case b = 0. The constant j =1 term (the sum of the terms not depending on N ) in the above expression is ln GK . These GK , b ≥ 1 are the generalized Glaisher–Kinkelin constants introduced by Bendersky, [5], and identiﬁed in the following closed form: b+1 (−1) B b+1 () ln GK = H − ζ (−b), for b ≥ 1 , b b b + 1 123 Khinchin Families and Hayman Class 893 by Choudhury [7], and Adamchik [1]. See also Wang [27]. The proper (original) Glaisher–Kinkelin constant is GK . By taking H = 0, formula () is also valid for 1 0 b = 0. The constant GK is actually 2π. With this notation, we have that, as s ↓ 0, 1 b + 1 b + 1 1 b −s ln W (e ) = ζ 1 + −ζ(−b) ln s+a ζ (−b)+o(1), (b+1)/a a a a s (6.12) where for compactness we have used that ζ(−b) = (−1) B /(b + 1),for b ≥ 0. b+1 6.3 Hayman Class and Coefficients of Partition Functions Let us apply Theorem 4.1 to check that these partition functions, P, Q, P and a,b W , are in the Hayman class and, thus, that they are strongly Gaussian. The variance and major arc conditions of Theorem 4.1 are obtained quite directly in each case. As usual, the minor arc estimates demand more attention. Recall that in Sect. 6.1 we have obtained, for these partition functions, suitable approximations of their mean and variance functions all satisfying condition (3.9) of Theorem C, and that we have just described in Sect. 6.2 appropriate asymptotics of these partition functions in the interval (0, 1). 6.3.1 Partition Function P. Theorem of Hardy–Ramanujan We start, as usual, with the case of the partition function P. Theorem 6.2 The partition function P is in the Hayman class. Proof We check that P satisﬁes the conditions of Theorem 4.1. For the variance function we have, see (6.3), that (3)ζ (2) 2 −s () σ (e ) ∼ , as s ↓ 0 . Thus, the variance condition of Theorem 4.1 is satisﬁed. g(z) We have that P(z) = e ,for z ∈ D, where g is the function ∞ ∞ jk k 1 z 1 z g(z) = ln = = , for z ∈ D j k 1 − z k k 1 − z j =1 j ,k≥1 k=1 (g has non-negative Taylor coefﬁcients, as the second expression above shows). Next we search for a cut function h so that conditions (4.3) and (4.4) of Theorem 4.1 are satisﬁed. We will take h(t ) = (1 − t ) ,for t ∈ (0, 1), for an appropriate α> 0to be speciﬁed shortly. Let F be the auxiliary function of P.Now, −s t g (t ) ≤ F (−s), for t = e and s > 0 , 123 894 A. Cantón et al. and so, because of the asymptotic formula (3.3), we have that t g (t ) = O , as t ↑ 1 . (1 − t ) Thus, if α> 4/3, the cut function h satisﬁes the major arc condition (4.3). ı θ For the minor arc, write for z = te , with t ∈ (0, 1) and θ ∈ R k k 1 z t ı θ Re(g(te )) − g(t ) = Re − . k k k 1 − z 1 − t k=1 All the summands in the series above are non- positive, so that keeping only the term corresponding to k = 1, we deduce that z t 1 1 ı θ Re(g(te )) − g(t ) ≤ Re − = Re − · 1 − z 1 − t 1 − z 1 − t Because of Lemma 4.3, we have that 1 1 1 1 ı θ sup Re(g(te )) − g(t ) ≤ Re − ≤ − · ıh(t ) ıh(t ) 1 − te 1 − t 1 − te 1 − t h(t )≤|θ |≤π And thus, from Lemma 4.4 we deduce, for some δ> 0 and some t ∈ (0, 1), that h(t ) ı θ sup Re(g(te )) − g(t ) ≤−δ , for t < t < 1 . (1 − t ) h(t )≤|θ |≤π Finally, from the moderate growth of σ(t ) given by (), we conclude that condi- tion (4.4) on the minor arc is amply satisﬁed as long as α< 3/2. In conclusion, by taking α so that 4/3 <α < 3/2, all the conditions of Theorem 4.1 are satisﬁed, and P is in the Hayman class. We can now deduce the asymptotic formula for p(n), the coefﬁcients of the partition −s 2 function P.From (6.2) and (6.3) we know that we can take m (e ) = ζ(2)/s and 2 −s 3 σ (e ) = (3)ζ (2)/s satisfying (3.9). −s Thus with τ = e and s = ζ(2)/n,for n ≥ 1, by appealing to formula (3.10) n n and the asymptotics (6.9)of P on (0, 1), we deduce that √ √ 1 1 π 2/3 n p(n) ∼ e , as n →∞ ; (6.13) 4 3 the Hardy–Ramanujan partition theorem, [13]. 123 Khinchin Families and Hayman Class 895 6.3.2 Function Q For the ogf Q of partitions with distinct parts given by ∞ ∞ ∞ n j Q(z) = q(n)z = (1 + z ) = , for z ∈ D , 2 j +1 1 − z n=0 j =1 j =0 we have Q = exp(g), where k+1 (−1) 1 kj k(2 j +1) g(z) = z = z , for z ∈ D . k k k, j ≥1 k≥1; j ≥0 The second expression shows that g has non-negative coefﬁcients. As in the case of P,wehave g (t ) = O(1/(1 − t ) ),as t ↑ 1. Thus, for a cut h(t ) = (1 − t ) , we just need α> 4/3 to fulﬁll condition (4.3) for the major arc of Theorem 4.1. ı θ Now, the minor arc. For z = te with t ∈ (0, 1) and θ ∈ R we have that ı θ 1 + te 2t (cos θ − 1) t (t /2(cos θ −1) (Rez−|z|)/2 = 1 + ≤ 1 + (cos θ − 1) ≤ e = e , 1 + t (1 + t ) 2 and, so, | Q(z)| 1 z t 1 1 1 ≤ exp Re − = exp Re − , Q(|z|) 4 1 − z 1 − t 4 1 − z 1 − t and, consequently, | Q(z)| 1 1 1 Reg(z) − g(|z|) = ln ≤ Re − . Q(|z|) 4 1 − z 1 − t With this and the same argument as in the case of P, we see that the cut function h(t ) = (1 − t ) with α< 3/2 satisﬁes the condition for the minor arc of Theorem 4.1. Thus, with α such that 4/3 <α < 3/2, we conclude that: Theorem 6.3 The partition function Q (of partitions with distinct parts) is in the Hayman class. The approximations m and σ for the mean and variance functions of the Khinchin family of Q given in formula (6.4), and the asymptotics (6.10)of Q in the interval (0, 1), plus strong gaussianity and Theorem C, allow us to derive the asymptotic formula for the number q(n) of partitions of n with distinct parts, see [13]: √ √ 1 1 π 1/3 n q(n) ∼ e , as n →∞ . 1/4 3/4 4 · 3 n 123 896 A. Cantón et al. 6.3.3 Function P : Theorem of Ingham a,b We consider now, for integers a, b ≥ 1, the ogf P (z) = , for z ∈ D , a,b aj +b 1 − z j =0 of partitions with parts in the arithmetic progression {aj + b : j ≥ 0}. If gcd(a, b) = d > 1, then P is not strongly Gaussian (although it is Gaussian), a,b since we can write P (z) = G(z ), where G is a holomorphic function in D, and a,b thus the coefﬁcients of P satisfy no asymptotic formula; see Remark 6.6 below. a,b We will verify by means of Theorem 4.1 that if gcd(a, b) = 1, then P is strongly a,b Gaussian. The condition that a and b are relatively prime is used exclusively to verify the minor arc condition of Theorem 4.1. We have already obtained, see formula (6.5), convenient approximations m and σ of the mean and variance functions of the Khinchin family of P , and an appropriate a,b asymptotic formula (6.11)of P on the interval (0, 1). a,b We can write P = exp(g), where a,b ∞ ∞ bk 1 1 z 1 k(aj +b) k g(z) = z = = U (z ), for z ∈ D , ak k k 1 − z k k=1 k=1 k≥1; j ≥0 where U is the rational function U (z) = , 1 − z holomorphic in D. Observe that the Taylor coefﬁcients of U around z = 0 are non- negative. We search now for a cut function h(t ) = (1 − t ) with appropriate α> 0. Since g (t ) = O(1/(1 − t ) ),as t ↑ 1, the condition (4.3) of Theorem 4.1 for the major arc is satisﬁed if α> 4/3. Next, the minor arc. This is where we use the condition gcd(a, b) = 1. We have k k Reg(z) − g(|z|) = ReU (z ) − U (|z| ) ≤ ReU (z) − U (|z|), for z ∈ D . k=1 The inequality holds since all the summands in the series above are non-positive. 2πıj /a The function U (z) has simple poles at the ath roots of unity: γ = e , with b+1 0 ≤ j < a. The residue of U at γ is −γ /a. ı φ Let us denote by R the region R ={z = re : 1/2 ≤ r < 1, |φ − 2π j /a|≤ j j a−1 π/a}, and let R = R ; observe that we do not include R in R. j 0 j =1 123 Khinchin Families and Hayman Class 897 For a certain constant M > 0 wehavethat b+1 () U (z) + ≤ M , for z ∈ R and 0 ≤ j < a . a(z − γ ) We start analyzing ReU (z) for z ∈ R. Lemma 6.4 For 0 < j < a, we have that 1 −|z| 1 b lim sup sup Re U (z) ≤ 1 + cos 2π j . |z| 2 a t ↑1 z∈R ; t ≤|z|≤1 Moreover, for certain δ ∈ (0, 1) and t ∈ (1/2, 1) we have that 1 −|z| Re U (z) ≤ δ, for every z ∈ R with |z| > t . |z| Proof Fix 0 < j < a and z ∈ R .Let w ∈ R such that z = γ w. By appealing j 0 j to () we have that a a a 1 −|z| 1 −|z| 1 −|w| 1 −|z| () Re U (z) ≤ Re γ + M . b b b |z| a(1 −|z|)|z| 1 − w |z| Now, for every w ∈ D one has that 1 −|w| 1 1 ∈ D , ∪{1} , 1 − w 2 2 and, in particular, 1 −|w| 1 ı η Re e ≤ (1 + cos(η)) , for all η ∈ R and w ∈ D . 1 − w 2 Therefore, we obtain from () that a a a 1 −|z| 1 −|z| 1 −|z| () Re U (z) ≤ δ + M . b b b |z| a(1 −|z|)|z| |z| with δ = (1 + cos(2πıjb/a)) . This bound () gives the lim sup of the statement. The bound now follows since gcd(a, b) = 1 and 0 < j < a imply that cos(2πıjb/a)< 1. As a consequence of Lemma 6.4,wehave 1 − δ () Re U (z) − U (t ) ≤ (δ − 1)U (t ) ≤− , for z ∈ R with |z|= t ∈ (t , 1). a 1 − t 123 898 A. Cantón et al. It remains to analyze ReU (z) in the region R .For z ∈ R , we have that 0 0 U (z) − ≤ M , a(1 − z) and, therefore, for z ∈ R with |z|= t, we have that 1 1 1 1 t Re U (z) − U (t ) ≤ M + Re − + − · a 1 − z 1 − t a(1 − t ) (1 − t ) Now, 1 t b − ≤ , for each t ∈ (0, 1), a(1 − t ) (1 − t ) a and thus from Lemmas 4.3 and 4.4, arguing as in the case above of P and Q, and increasing the t above if necessary, we deduce that for certain δ> 0, b h(t ) () sup (Re U (z) − U (t )) ≤ M + − δ , for t ∈ (t , 1). ı θ a (1 − t ) z=te ∈R ; h(t )≤|θ | The minor arc for {|z|= t } comprises R ∩{|z|= t } (where we have the bound ()) ı θ and the two subarcs in R ∩{|z|= t } of those z = te where |θ|≥ h(t ). From () and () and the moderate growth of σ,see (6.5), we conclude thatα< 3/2 sufﬁces to guarantee the minor arc condition of Theorem 4.1, and consequently that: Theorem 6.5 If gcd(a, b) = 1,P is in the Hayman class. a,b The approximations m and σ for the mean and variance functions of the Khinchin family of P given in (6.5) and the asymptotics (6.11)of P in the interval (0, 1), a,b a,b plus strong gaussianity and Theorem C, allow us to derive Ingham’s theorem, originally in [17], seealsoKane [18]: for integers a, b ≥ 1 with gcd(a, b) = 1, the number of partitions p (n) of n ≥ 1 in parts drawn for the arithmetic sequence {aj + b : j ≥ 0} a,b satisﬁes that, as n →∞, % & b/(2a) 1 b π 1 2 b/(2a)−1/2 p (n)∼ √ a exp π n . a,b 1/2+b/(2a) a 6 n 3a 22π (6.14) Remark 6.6 If gcd(a, b) = d > 1, then we have P (z) = P (z ), for z ∈ D , a,b a ,b 123 Khinchin Families and Hayman Class 899 where a = a/d and b = b/d. Observe that gcd(a , b ) = 1. We deduce from (6.14) that, as n →∞, coef P = coef P [nd] a,b [n] a ,b % & b/(2a) 1 b π b/(2a)−1/2 ∼ d √ a a 6 22π 1 2 × exp π nd . 1/2+b/(2a) (nd) 3a All the coefﬁcients of P (z) whose indices are not multiples of d are 0. a,b 6.3.4 Plane and Colored Partitions: Wright’s Theorem We deal here with W , where b ≥ 0, the ogf of colored partitions with coloring sequence ( j ) . j ≥1 First we discuss the case W and then we describe the minor changes to approach the general case b ≥ 1. a) Plane partitions. The MacMahon function ∞ ∞ 1 n W (z) = M z 1 − z j =1 n=0 codiﬁes plane partitions: M is the number of plane partitions of the integer n,for n ≥ 1, with M = 1. We have already obtained, see (6.6) and (6.7), convenient approximations, m and σ , of the mean and variance functions of the Khinchin family of W : 1 1 −s 2 −s m (e ) = ζ(3)(3) and σ (e ) = ζ(3)(4) , as s ↓ 0 , 3 4 s s which satisfy condition (3.9). We also have at our disposal, see equation (6.12), a suitable asymptotic formula for W on the interval (0, 1): 1/12 2 e 2 1 −s ζ (−1) 1/12 ζ(3)/s 1/12 ζ(3)/s W (e ) ∼ e s e = s e , as s ↓ 0 . (6.15) GK We apply now Theorem 4.1 to verify that W is strongly Gaussian. The variance condition is clearly satisﬁed. We look now for a cut function h(t ) = (1 − t ) , with α> 0 to be speciﬁed. Let g(z) = ln W (z) in D.Wehave j 1 l j l g(z) = (z ) = K (z ), for z ∈ D , l l l, j ≥1 l=1 123 900 A. Cantón et al. where K is the Koebe function: z 1 1 + z 1 K (z) = = − , for z ∈ D . (1 − z) 4 1 − z 4 Observe, in particular, that the coefﬁcients of g are all non-negative. As g (t ) = O(1/(1 − t ) ),as t ↑ 1, see (3.4), the condition on the major arc (4.3) is readily satisﬁed if α> 5/3. ı θ Now, the minor arc. For z = te with t ∈ (0, 1) and θ ∈ R, we have that l l Reg(z) − g(t ) = Re K (z ) − K (t ) ≤ Re K (z) − K (t ) l=1 2 2 1 1 + z 1 + t = Re − , 4 1 − z 1 − z since the summands above are all non-positive, and, thus, that 2 2 1 1 + z 1 + t 1 1 1 Reg(z) − g(t ) ≤ − ≤ (1 + t ) − 2 2 4 1 − z 1 − t 4 |1 − z| (1 − t ) 1 1 1 1 1 1 − t ≤ − = − 1 . 2 2 2 ı θ 4 |1 − z| (1 − t ) 4 (1 − t ) 1 − te Appealing to Lemma 4.4 we deduce that for t ≥ t and some δ> 0 we have that h(t ) sup (Reg(z) − g(t )) ≤−δ , (1 − t ) |θ |≥h(t ) and conclude that condition (4.4) on the minor arc is satisﬁed whenever α< 2. Thus, specifying α ∈ (5/3, 2), we conclude that: Theorem 6.7 The MacMahon function W is in the Hayman class. −s 1/3 For each n ≥ 1, take τ = e with s = (ζ (3)(3)/n) , so that m (τ ) = n. n n n Plugging τ in the asymptotic formula (3.10) and using (6.15) we obtain % & ζ (−1) e 1 7/36 25/36 1/3 2/3 M ∼ √ ζ(3) 2 exp 3(ζ (3)/4) n , as n →∞ , 25/36 12π (6.16) which is Wright’s asymptotic formula for plane partitions, [29]. b) Colored partitions. Fix an integer b ≥ 1. The argument which follows to han- b 1 dle W is just an extension of the one above for plane partitions, W ;ﬁrst,strong 1 1 gaussianity via Theorem 4.1 and then asymptotics of coefﬁcients via Theorem C. Theorem 6.8 For b ≥ 1, the function W is in the Hayman class. 123 Khinchin Families and Hayman Class 901 Proof Introduce the power series Q and R given by b b ∞ ∞ j ! w b j j Q (w) = j w and R (w) = w = b! , b b b+1 ( j − b)! (1 − w) j =1 j =b and observe that ∞ ∞ 1 1 b b kj k g(z) = ln W (z) = j z = Q (z ). k k j ,k k=1 ı θ For z = te ∈ D, we have that Reg(z) − g(t ) ≤ Re Q (z) − Q (t ) b b b b z t 1 1 ≤ b! Re − ≤ b! t − . b+1 b+1 b+1 b+1 (1 − z) (1 − t ) |1 − z| (1 − t ) For a potential cut function h(t ) = (1 − t ) , with α> 0, we have, because of Lemmas 4.3 and 4.4, that for some δ > 0, (b+1)+2 (1 − t ) ı θ lim sup sup Reg(te ) − g(t ) ≤−δ < 0 . h(t ) t ↑1 h(t )≤|θ |≤π Thus, condition (4.4) on the minor arc of Theorem 4.1 is satisﬁed as long as α< 1 + (b + 1)/2. (b+1)+3 From (3.4), we deduce that g (t ) = O(1/(1 − t ) ),as t ↑ 1, and, thus, that condition on the major arc is satisﬁed whenever α> 1 + (b + 1)/3. Now from the approximation of moments of the Khinchin family of W given by (6.6), (6.7) and (6.8), and the asymptotics of W on the interval (0, 1), we deduce from Theorem C the following asymptotic formula for the coefﬁcients a of W : 1 (b+1)/(b+2) γ n a ∼ α e , as n →∞ , (6.17) n b where ζ (−b) (−2ζ(−b)+1)/(2(b+2)) α = √ e √ [(b + 2)ζ(b + 2)] , 2π b + 2 −2ζ(−b) + b + 3 b + 2 1/(b+2) β = ,γ = ((b + 2)ζ(b + 2)) . b b 2(b + 2) b + 1 Formula (6.17) reduces to (6.13) when b = 0, usual partitions, and to (6.16) when b = 1, plane partitions. 123 902 A. Cantón et al. 6.3.5 General W For a ≥ 1, b ≥ 0, general, the ogfs W (z) are all Gaussian. We do have appropriate approximations of the mean and variance functions given by (6.6), (6.7) and (6.8), satisfying condition (3.9) towards applying Theorem C, and also a convenient asymptotic formula in the interval (0, 1) given by (6.12). Besides, aiming at applying Theorem 4.1 and verifying the strong gaussianity b α of W , for a potential cut function h(t ) = (1 − t ) , with α> 0, we do have, because of (3.4), that condition (4.3) on the major arc is satisﬁed as long as α> 1 + (b + 1)/(3a). But, alas, we have not been able to stretch the simple estimates above, for the case a = 1, to verify the minor arc condition (4.4), which should be satisﬁed if α< 1 + (b + 1)/(2a). If that were the case, all the ogfs W would be in the Hayman class, and theefore they would be strongly Gaussian. And from all the above, we would have the following asymptotic formula for the coefﬁcients a of W : 1 (b+1)/(b+1+a) γ n a,b a ∼ α e , as n →∞, n a,b a,b where 1 a (a−2aζ(−b))/(2(a+b+1)) aζ (−b) α = √ e C , a,b a,b a + b + 1 2π −2aζ(−b) + 2a + b + 1 a + b + 1 a/(a+b+1) β = ,γ = C , a,b a,b a,b 2(a + b + 1) b + 1 with 1 b + 1 b + 1 C = ζ 1 + 1 + . a,b a a a For asymptotic formulae (and further asymptotic expansions) in the cases W and W , of partitions into squares and ath powers, obtained via the circle method, we refer to Vaughan [26], Gafni [10] and the primordial Wright [31]. It would be nice to know whether the ogf W , with a ≥ 2, of partitions into powers of a is in the Hayman class. Acknowledgements We are extremely grateful to the referees of an earlier version of this paper for their insightful and valuable suggestions and recommendations. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted 123 Khinchin Families and Hayman Class 903 by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Adamchik, V.S.: Polygamma functions of negative order. J. Comput. Appl. 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Ramanujan J. 11(1), 49–66 (2006) 19. Macintyre, A.J., Wilson, R.: Operational methods and the coefﬁcients of certain power series. Math. Ann. 127, 243–250 (1954) 20. MacMahon, P.A.: Combinatory Analysis, vol. 2. Cambridge University Press, Cambridge (1916) 21. Meinardus, G.: Asymptotische Aussage über Partitionen. Math. Z. 59, 388–398 (1953) 22. Moser, L., Wyman, M.: An asymptotic formula for the Bell numbers. Trans. R. Soc. Can. Sect. III(49), 49–54 (1955) 23. Moser, L., Wyman, M.: On solutions of x = 1 in symmetric groups. Can. J. Math. 7, 159–168 (1955) 24. Moser, L., Wyman, M.: Asymptotic expansions. Can. J. Math. 8, 225–233 (1956) 25. Rosenbloom, P.C.: Probability and entire functions. In: Szegö, G., et al. (eds.) Studies in mathematical analysis and related topics. Essays in honor of George Pólya, pp. 325–332. Stanford University Press, Stanford (1962) 26. Vaughan, R.C.: Squares: additive questions and partitions. Int. J. Number Theory 11(5), 1367–1409 (2015) 27. Wang, W.: Some asymptotic expansions on hyperfactorial functions and generalized Glaisher–Kinkelin constants. Ramanujan J. 43(3), 513–533 (2017) P(z) 28. Wilf, H.: The asymptotics of e and the number of elements of each order in S . Bull. Am. Math. Soc. (N.S.) 15(2), 228–232 (1986) 29. Wright, E.M.: Asymptotic partition formulae I. Plane partitions. Q. J. Math. 1, 177–189 (1931) 30. Wright, E.M.: The coefﬁcients of a certain power series. J. Lond. Math. Soc. 7(4), 256–262 (1932) 31. Wright, E.M.: Asymptotic partition formulae. III. Partitions into k-th powers. Acta Math. 63, 143–191 (1934) 123 904 A. Cantón et al. 32. Wright, E.M.: On the coefﬁcients of power series having exponential singularities. II. J. Lond. Math. Soc. 24, 304–309 (1949) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. Authors and Aﬃliations 1 2 2 3 Alicia Cantón · José L. Fernández · Pablo Fernández · Víctor J. Maciá José L. Fernández joseluis.fernandez@uam.es Alicia Cantón alicia.canton@upm.es Pablo Fernández pablo.fernandez@uam.es Víctor J. Maciá victor.macia@wustl.edu Departamento de Matemática e Informática Aplicadas a las Ingenierías Civil y Naval, ETSIN, Universidad Politécnica de Madrid, Avenida de la Memoria 4, Ciudad Universitaria, 28040 Madrid, Spain Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco s/n, 28049 Madrid, Spain Department of Mathematics and Statistics, Washington University in St Louis, 1 Brookings Drive, St. Louis, MO 63130, USA

Journal

Computational Methods and Function Theory – Springer Journals

Published: Dec 1, 2021

Keywords: Khinchin families; Hayman admissible functions; Asymptotic formulae; Partitions; Analytic combinatorics; Primary 30B10; Secondary 05A16; 11P82; 60F99

References

Polygamma functions of negative order

Adamchik, VS
Hardy–Ramanujan’s asymptotic formula for partitions and the central limit theorem

Báez-Duarte, L
General Tauberian theorems on the real line

Báez-Duarte, L
Enumeration of plane partitions

Bender, EA; Knuth, DE
Sur la fonction gamma généralisée

Bendersky, L
Une étude asymptotique probabiliste des coefficients d’une série entière

Candelpergher, B; Miniconi, M
The Riemann zeta-function and its derivatives

Choudhury, BK
Power partitions

Gafni, A
Developments in the Khintchine–Meinardus probabilistic method for asymptotic enumeration

Granovsky, BL; Stark, D
Meinardus’ theorem on weighted partitions: extensions and a probabilistic proof

Granovsky, BL; Stark, D; Erlihson, M
Asymptotic formulae in combinatory analysis

Hardy, GH; Ramanujan, SA
The number of idempotent elements in symmetric semigroups

Harris, B; Schoenfeld, L
Asymptotic expansions for the coefficients of analytic functions

Harris, B; Schoenfeld, L
A generalisation of Stirling’s formula

Hayman, WK
A Tauberian theorem for partitions

Ingham, AE
An elementary derivation of the asymptotics of partition functions

Kane, DM
Operational methods and the coefficients of certain power series

Macintyre, AJ; Wilson, R
Asymptotische Aussage über Partitionen

Meinardus, G
An asymptotic formula for the Bell numbers

Moser, L; Wyman, M
On solutions of xd=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^d=1$$\end{document} in symmetric groups

Moser, L; Wyman, M
Asymptotic expansions

Moser, L; Wyman, M
Squares: additive questions and partitions

Vaughan, RC
Some asymptotic expansions on hyperfactorial functions and generalized Glaisher–Kinkelin constants

Wang, W
The asymptotics of eP(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{P(z)}$$\end{document} and the number of elements of each order in Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n$$\end{document}

Wilf, H
Asymptotic partition formulae I. Plane partitions

Wright, EM
The coefficients of a certain power series

Wright, EM
Asymptotic partition formulae. III. Partitions into k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-th powers

Wright, EM
On the coefficients of power series having exponential singularities. II

Wright, EM

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Khinchin Families and Hayman Class

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Khinchin Families and Hayman Class

Khinchin Families and Hayman Class

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