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Sums of finite products of Bernoulli functions

Sums of finite products of Bernoulli functions Kwangwoon University, Seoul, In this paper, we consider three types of functions given by sums of finite products of 139-701, Republic of Korea Department of Mathematics, Bernoulli functions and derive their Fourier series expansions. In addition, we express College of Science, Tianjin each of them in terms of Bernoulli functions. Polytechnic University, Tianjin, 300160, China MSC: 11B68; 42A16 Full list of author information is available at the end of the article Keywords: Fourier series; sums of finite products of Bernoulli functions 1 Introduction As is well known, the Bernoulli polynomials B (x) are given by the generating function t t xt e = B (x) (see [–]). (.) e – m! m= When x =, B = B () are called Bernoulli numbers. For any real number x,welet m m x = x –[x] ∈ [, ) (.) denote the fractional part of x. Fourier series expansion of higher-order Bernoulli functions was treated in the recent paper []. Here we will consider the following three types of functions given by sums of finite products of Bernoulli functions and derive their Fourier series expansions. In addition, we will express each of them in terms of Bernoulli functions. () α (x)= B (x)B (x) ··· B (x) (m ≥ ); m c c c   r c +c +···+c =m,c ,...,c ≥   r  r () β (x) = B (x)B (x) ··· B (x) (m ≥ ); m c c c c +c +···+c =m,c ,...,c ≥   r   r  r c !c !···c !   r () γ (x)= B (x)B (x) ··· B (x) (m ≥ r). r,m c c c c +c +···+c =m,c ,...,c ≥   r   r  r c c ···c   r For elementary facts about Fourier analysis, the reader may refer to any book (for example, see [, ]). As to β (x), we note that the next polynomial identity follows immediately from The- orems . and ., which is in turn derived from the Fourier series expansion of β (x): j–   r B (x)B (x) ··· B (x)=  +  B (x), c c c m+ m–j+ j   c !c ! ··· c ! r j!   r c +c +···+c =m   r j= © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 2 of 15 where r B B ··· B c c c   a = .(.) a c !c ! ··· c !   a max{,r–l}≤a≤r– c +c +···+c =l+a–r   a The obvious polynomial identities can be derived also for α (x)and γ (x)from The- m m orems . and .,and Theorems . and ., respectively. It is remarkable that from the m– Fourier series expansion of the function B (x)B (x)we can derive the k m–k k= k(m–k) Faber-Pandharipande-Zagier identity (see [–]) and the Miki identity (see [–]). 2 The function α (x) Let α (x)= B (x)B (x) ··· B (x)(m ≥ ). Here the sum runs over all non- m c c c c +c +···+c =m   r   r negative integers c , c ,..., c with c + c + ··· + c = m (r ≥ ). Then we will consider the   r   r function α x = B x B x ··· B x,(.) m c c c   c +c +···+c =m   r defined on (–∞, ∞), which is periodic with period . The Fourier series of α (x)is (m) πinx A e , (.) n=–∞ where (m) –πinx A = α x e dx –πinx = α (x)e dx.(.) Before proceeding further, we need to observe the following. α (x)= c B (x)B (x) ··· B (x)  c – c c m r   c +c +···+c =m   r + ··· + c B (x)B (x) ··· B (x)B (x) r c c c c –   r– r = c B (x)B (x) ··· B (x)  c – c c   r c +c +···+c =m,c ≥   r  + ··· + c B (x)B (x) ··· B (x) r c – c c   r c +c +···+c =m,c ≥ r r   =(m + r –) B (x)B (x) ··· B (x) c c c   c +c +···+c =m–   =(m + r –)α (x). (.) m– From this, we have α (x) m+ = α (x)(.) m + r Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 3 of 15 and α (x) dx = α () – α ().(.) m m+ m+ m + r For m ≥ , we put = α () – α () m m m = B ()B () ··· B () – B B ··· B c c c c c c   r   r c +c +···+c =m   = (B + δ ) ··· (B + δ )– B B ··· B c ,c c ,c c c c r r r     c +c +···+c =m   r = B B ··· B – B B ··· B c c c c c c   a   r ≤a≤r c +c +···+c =m+a–r c +c +···+c =m   a   r a≥r–m = B B ··· B,(.) c c c   a max{,r–m}≤a≤r– c +c +···+c =m+a–r   a where we understand that, for r – m ≤ and a = , the inner sum is δ . m,r Observe here that the sum over all c + c + ··· + c = m of any term with a of B and b   r c of δ ( ≤ e, f ≤ r, a + b = r), all give the same sum ,c B ··· B δ ··· δ c c ,c ,c  a a+ a+b c +c +···+c =m   = B B ··· B,(.) c c c   c +c +···+c =m+a–r   a which is not an empty sum as long as m + a – r ≥ , i.e., a ≥ r – m. Thus α () = α () ⇐⇒  = (.) m m m and α (x) dx = .(.) m m+ m + r (m) Now, we are ready to determine the Fourier coefficients A . Case : n =. (m) –πinx A = α (x)e dx   –πinx  –πinx =– α (x)e + α (x)e dx πin πin m + r –  (m–) = A – πin πin m + r – m + r –   (m–) = A –  – m– m πin πin πin πin Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 4 of 15 (m + r –) (m + r –)  j– (m–) = A – m–j+  j (πin) (πin) j= = ··· (m + r –) (m + r –) m j– () = A – m–j+ m j (πin) (πin) j=  (m + r) =– ,(.) m–j+ m + r (πin) j= ()  –πinx where A = e dx =. Case : n =. (m) A = α (x) dx = .(.) m m+ m + r Let us recall the following facts about Bernoulli functions B (x): (a) for m ≥ , πinx B x =–m! ; (.) (πin) n=–∞ n = (b) for m =, πinx ⎨ B (x) for x ∈/ Z, – = (.) πin  for x ∈ Z. n=–∞ n = α (x)(m ≥ ) is piecewise C .Moreover, α (x) is continuous for those positive m m integers m with  =  and discontinuous with jump discontinuities at integers for those positive integers m with  =. Assume first that m is a positive integer with  =.Then α () = α (). Hence α (x) m m m m is piecewise C and continuous. Thus the Fourier series of α (x) converges uniformly to α (x), and α x = m m+ m + r ∞ m (m + r) πinx + –  e m–j+ m + r (πin) n=–∞,n = j= m ∞ πinx   m + r e =  +  –j! m+ m–j+ m + r m + r j (πin) n=–∞ j= n =   m + r =  +  B x m+ m–j+ j m + r m + r j j= B (x)for x ∈/ Z, +  × (.) for x ∈ Z. Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 5 of 15 We can now state our first result. Theorem . For each positive integer l, we let = B B ··· B . l c c c   max{,r–l}≤a≤r– c +c +···+c =l+a–r   a Assume that  = for a positive integer m. Then we have the following. (a) B (x)B (x) ··· B (x) has the Fourier series expansion c c c c +c +···+c =m     r B x B x ··· B x c c c   c +c +···+c =m   r ∞ m (m + r)   πinx =  –  e , m+ m–j+ m + r m + r (πin) n=–∞ j= n = for all x ∈ R, where the convergence is uniform. (b) B x B x ··· B x c c c   r c +c +···+c =m   r   m + r =  +  B x , m+ m–j+ j m + r m + r j j= for all x ∈ R, where B (x) is the Bernoulli function. Assume next that  =  for a positive integer m.Then α () = α (). Hence α (x)is m m m m piecewise C and discontinuous with jump discontinuities at integers. The Fourier series of α (x)converges pointwise to α (x)for x ∈/ Z and converges to m m   α () + α () = α () +  (.) m m m m   for x ∈ Z. Now, we can state our second result. Theorem . For each positive integer l, we let = B B ··· B . l c c c   a max{,r–l}≤a≤r– c +c +···+c =l+a–r   Assume that  = for a positive integer m. Then we have the following. ∞ m   (m + r) πinx (a)  + –  e m+ m–j+ m + r m + r (πin) n=–∞ j= n = B (x)B (x) ··· B (x) for x ∈/ Z, c c c c +c +···+c =m   r   ⎩  B B ··· B +  for x ∈ Z. c c c m c +c +···+c =m   r   r  Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 6 of 15   m + r (b)  +  B x m+ m–j+ j m + r m + r j j= = B x B x ··· B x for x ∈/ Z; c c c   r c +c +···+c =m   r   m + r +  B x m+ m–j+ j m + r m + r j j= = B B ··· B +  for x ∈ Z. c c c m   r c +c +···+c =m   r 3 The function β (x) Let β (x)= B (x)B (x) ··· B (x)(m ≥ ).Herethe sumrunsoverall m c c c c +c +···+c =m   r   r c !c !···c !   r nonnegative integers c , c ,..., c with c + c + ··· + c = m (r ≥ ). Then we will consider   r   r the function β x = B x B x ··· B x,(.) m c c c   r c !c ! ··· c !   r c +c +···+c =m   defined on (–∞, ∞), which is periodic with period . The Fourier series of β (x)is (m) πinx B e,(.) n=–∞ where   (m) –πinx –πinx B = β x e dx = β (x)e dx. (.) m m   Before proceeding further, we need to observe the following. β (x)= B (x)B (x) ··· B (x) c – c c m   r c !c ! ··· c !   r c +c +···+c =m   + ··· + B (x)B (x) ··· B (x) c c c –   c !c ! ··· c !   r = B (x)B (x) ··· B (x) c – c c   r (c –)!c ! ··· c !   r c +c +···+c =m,c ≥   r  + ··· + B (x)B (x) ··· B (x) c c c –   r c !c ! ··· (c –)!   r c +c +···+c =m,c ≥   r r = r B (x)B (x) ··· B (x) c c c   c !c ! ··· c !   r c +c +···+c =m–   r = rβ (x). (.) m– From this, we have β (x) m+ = β (x)(.) r Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 7 of 15 and β (x) dx = β () – β ().(.) m m+ m+ Let = β () – β () m m m B ()B () ··· B () B B ··· B c c c c c c   r   r = – c !c ! ··· c ! c !c ! ··· c !   r   r c +c +···+c =m c +c +···+c =m   r   r (B + δ )(B + δ ) ··· (B + δ ) c ,c c ,c c ,c r r     c !c ! ··· c !   r c +c +···+c =m   r B B ··· B c c c   r c !c ! ··· c !   r c +c +···+c =m   r r B B ··· B c c c   a = ,(.) a c !c ! ··· c !   a c +c +···+c =m+a–r max{,r–m}≤a≤r– a   where we understand that, for r – m ≤ and a = , the inner sum is δ . m,r Observe here that the sum over all c + c + ··· + c = m of any term with a of B and b   r c of δ ( ≤ e, f ≤ r, a + b = r), all give the same sum ,c B ··· B δ ··· δ c c ,c ,c  a+ a+b c !c ! ··· c !   r c +c +···+c =m   r B B ··· B c c c   = ,(.) c !c ! ··· c !   r c +c +···+c =m+a–r   a which is not an empty sum as long as m + a – r ≥ , i.e., a ≥ r – m. Also, we have β () = β () ⇔  = (.) m m m and β (x) dx = .(.) m m+ (m) Now, we would like to determine the Fourier coefficients B . Case : n =. (m) –πinx B = β (x)e dx   –πinx  –πinx =– β (x)e + β (x)e dx πin πin  r –πinx =– β () – β () + β (x)e dx m m m– πin πin  Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 8 of 15 r  (m–) = B – πin πin r r   (m–) = B –  – m– m πin πin πin πin j– r r (m–) = B – m–j+ πin (πin) j= = ··· m m j– r r () = B – m–j+ πin (πin) j= j– =– ,(.) m–j+ (πin) j= ()  –πinx where B = e dx =. Case : n =. (m) B = β (x)= .(.) m m+ β (x)(m ≥ ) is piecewise C .Moreover, β (x) is continuous for those positive m m integers m with  =  and discontinuous with jump discontinuities at integers for those positive integers m with  =. Assume first that  =  for a positive integer m.Then β () = β (). Hence β (x)is m m m m piecewise C and continuous. Thus the Fourier series of β (x) converges uniformly to β (x), and ∞ m j–  r πinx β x =  + –  e m m+ m–j+ r (πin) n=–∞ j= n = m ∞ j– πinx  r e =  +  × –j! m+ m–j+ r j! (πin) n=–∞ j= n = j–  r =  +  B x m+ m–j+ j r j! j= B (x)for x ∈/ Z, +  × (.) for x ∈ Z. Now, we can state our first result. Theorem . For each positive integer l, we let r B B ··· B c c c   = .(.) a c !c ! ··· c !   a max{,r–l}≤a≤r– c +c +···+c =l+a–r   a Assume that  = for a positive integer m. Then we have the following. m Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 9 of 15 (a) B (x)B (x) ··· B (x) has the Fourier series expansion c c c c +c +···+c =m   r r c !c !···c !     r B x B x ··· B x c c c   r c !c ! ··· c !   r c +c +···+c =m   r ∞ m j–  r πinx =  –  e , (.) m+ m–j+ r (πin) n=–∞ j= n = for all x ∈ (–∞, ∞), where the convergence is uniform. (b) B x B x ··· B x c c c   c !c ! ··· c !   r c +c +···+c =m   r j–  r =  +  B x,(.) m+ m–j+ j r j! j= for all x ∈ (–∞, ∞), where B (x) is the Bernoulli function. Assume next that m is a positive integer with  =.Then β () = β (). Hence β (x) m m m m is piecewise C and discontinuous with jump discontinuities at integers. Thus the Fourier series of β (x)converges pointwise to β (x)for x ∈/ Z and converges to m m   β () + β () = β () + m m m m     = B B ··· B +  (.) c c c m   r c !c ! ··· c !    r c +c +···+c =m   r for x ∈ Z. Now, we can state our second result. Theorem . For each positive integer l, let r B B ··· B c c c   = .(.) a c !c ! ··· c !   a max{,r–l}≤a≤r– c +c +···+c =l+a–r   a Assume that  = for a positive integer m. Then we have the following. ∞ m j–  r πinx (a)  –  e m+ m–j+ r (πin) n=–∞,n = j= ⎨  B (x)B (x) ··· B (x) for x ∈/ Z, c c c c +c +···+c =m c !c !···c !     r r     B B ··· B +  for x ∈ Z. c c c m c +c +···+c =m c !c !···c !      r r   Here the convergence is pointwise. j–  r (b)  +  B x m+ m–j+ j r j! j= = B x B x ··· B x for x ∈/ Z, c c c   r c !c ! ··· c !   r c +c +···+c =m   r Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 10 of 15 j–  r +  B x m+ m–j+ j r j! j=   = B B ··· B +  for x ∈ Z. c c c m   r c !c ! ··· c !    r c +c +···+c =m   r Here B (x) is the Bernoulli function. 4 The function γ (x) r,m Let γ (x)= B (x)B (x) ··· B (x)(m ≥ r ≥ ). Here the sum r,m c c c c +c +···+c =m,c ,...,c ≥ c c ···c     r  r r   is over all positive integers c , c ,..., c with c + c + ··· + c = m.   r   r γ (x)= B (x)B (x) ··· B (x) c – c c r,m   r c ··· c  r c +c +···+c =m,c ,...,c ≥   r  r + B (x)B (x) ··· B (x) c c – c   r c c ··· c   r c +c +···+c =m,c ,...,c ≥   r  r + ··· + B (x)B (x) ··· B (x) c c c –   r c c ··· c   r– c +c +···+c =m,c ,...,c ≥   r  r = B (x) ··· B (x) c c  r c ··· c  r c +···+c =m–,c ,...,c ≥  r  r + B (x) ··· B (x) c c  r c ··· c  r c +···+c =m–,c ,...,c ≥  r  r + ··· + B (x) ··· B (x) c c  r– c c ··· c   r– c +c +···+c =m–,c ,...,c ≥   r–  r– + B (x) ··· B (x) c c c c ··· c   r– c +c +···+c =m–,c ,...,c ≥ r r    = rγ (x)+(m –)γ (x). (.) r–,m– r,m– Thus, γ (x)= rγ (x)+(m –)γ (x)(m ≥ r), (.) r–,m– r,m– r,m with γ (x)=. r,r– Replacing m by m +, we get mγ (x)= γ (x)– rγ (x). (.) r,m r–,m r,m+ Denoting γ (x) dx by a ,wehave r,m r,m r  a =– a + , (.) r,m r–,m r,m+ m m where = γ () – γ (). From the recurrence relation (.), we can easily show that r,m r,m r,m r– (r) j– j– γ (x) dx = (–) ,(.) r,m r–j+,m+ j= Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 11 of 15 = γ () – γ () r,m r,m r,m B () ··· B () c c  r c ··· c  r c +c +···+c =m,c ,...,c ≥   r  r B ··· B c c c ··· c  r c +c +···+c =m,c ,...,c ≥   r  r = (B + δ ) ··· (B + δ ) c ,c c ,c   r r c +c +···+c =m,c ,...,c ≥   r  r B ··· B c c c ··· c  r c +c +···+c =m,c ,...,c ≥ r r    r B B ··· B c c c   = . (.) a c c ··· c   a ≤a≤r– c +c +···+c =m+a–r,c ,...,c ≥   a  a Observe here that the sum over all positive integers c ,..., c satisfying c + c + ··· + c =  r   r m of any term with a of B and b of δ ( ≤ e, f ≤ r, a + b = r), all give the same sum c ,c B ··· B δ ··· δ c c ,c ,c  a a+ a+b c c ··· c   r c +c +···+c =m,c ,...,c ≥   r  a B B ··· B c c c   a = ,(.) c c ··· c   a c +c +···+c =m+a–r,c ,...,c ≥   a  a and that, as m + a – r ≥ a,there arenoempty sums. Here we note that, for a = , the inner sum is δ since it corresponds to the sums m,r δ δ ··· δ ,c ,c ,c   r .(.) c c ··· c   r c +c +···+c =m,c ,...,c ≥   r  r Also, γ () = γ () ⇔ =. r,m r,m r,m Now, we would like to consider the function γ x = B x B x ··· B x,(.) r,m c c c   c c ··· c   r c +c +···+c =m,c ,...,c ≥ r r    defined on (–∞, ∞), which is periodic with period . The Fourier series of γ (x)is r,m (r,m) πinx C e,(.) n=–∞ where   (r,m) –πinx –πinx C = γ x e dx = γ (x)e dx.(.) r,m r,m   (r,m) Now, we are going to determine the Fourier coefficients C . n Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 12 of 15 Case : n =. (r,m) –πinx C = γ (x)e dx r,m   –πinx  –πinx =– γ (x)e + γ (x)e dx r,m r,m πin πin   =– γ () – γ () + rγ (x)+(m –)γ (x) r,m r,m r–,m– r,m– πin πin –πinx × e dx   r (r,m–) (r–,m–) =– + C + C .(.) r,m n n πin πin πin From this, we obtain m – r  (r,m) (r,m–) (r–,m–) C = C + C – r,m n n n πin πin πin m – m – r  (r,m–) (r–,m–) = C + C – r,m– n n πin πin πin πin r  (r–,m–) + C – r,m πin πin   (m –) r(m –) (m –) j– j– (r,m–) (r–,m–j) = C + C – r,m–j+ n n  j j (πin) (πin) (πin) j= j= = ··· m–r m–r (m –) r(m –) (m –) m–r j– j– (r,r) (r–,m–j) = C + C – . (.) r,m–j+ n n m–r j j (πin) (πin) (πin) j= j= Here,  r (r,r) –πinx C = x – e dx r  r–   r  –πinx –πinx =– x – e + x – e dx πin  πin  r r    r (r–,r–) =– – – + C,(.) πin   πin and r r   = γ () – γ () = – – . (.) r,r r,r r,r   Thus  r (r,r) (r–,r–) C =– + C.(.) r,r n n πin πin Finally, we obtain, for n =, m–r+ m–r+ r(m –) (m –) j– j– (r,m) (r–,m–j) C = C – .(.) r,m–j+ n n j j (πin) (πin) j= j= Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 13 of 15 Also, we note that, for n =,  (m –)! (,m) –πinx C = B (x)e dx =–.(.) m (πin) (r,m) Thus, for n =, (.)togetherwith(.)determine all C recursively. Case : n =. (r) j– (r,m) j– C = γ (x) dx = (–) .(.) r,m r–j+,m+ j= γ (x)(m ≥ r ≥ ) is piecewise C . In addition, γ (x) is continuous for those pos- r,m r,m itive integers r, m with =  and discontinuous with jump discontinuities at integers r,m for those positive integers r, m with =. r,m Assume first that = for some integers r, m with m ≥ r ≥ . Then γ () = γ (). r,m r,m r,m Hence γ (x)is piecewise C and continuous. Thus the Fourier series of γ (x)con- r,m m verges uniformly to γ (x), and (r,m) (r,m) πinx γ x = C + C e ,  n n=–∞ n = (r,m) (r,m) where C is given by (.), and C ,for each n = , are determined by relations (.) and (.). Now, we are ready to state our first theorem. Theorem . For all integers s, l, with l ≥ s ≥ , we let s B ··· B c c  a s,l a c ··· c  a ≤a≤s– c +···+c =l+a–s,c ,...,c ≥  a  a s B ··· B c c  a = δ + .(.) s,l a c ··· c  a ≤a≤s– c +c +···+c =l+a–s,c ,...,c ≥   a  a Assume that = for some integers r,mwithm ≥ r ≥ . Then we have the following. r,m B (x) ··· B (x) has the Fourier series expansion c c  r c +c +···+c =m,c ,...,c ≥ c ···c   r  r  r B x ··· B x c c c ··· c  r c +c +···+c =m,c ,...,c ≥ r r    (r,m) (r,m) πinx = C + C e ,  n n=–∞,n = (r) (r,m) (,m) (r,m) r– j– j– where C = (–) , with C =, and C , for each n =, are de- j r–j+,m+ n  j=  termined recursively from m–r+ m–r+ r(m –) (m –) j– j– (r,m) (r–,m–j) C = C – ,(.) r,m–j+ n n j j (πin) (πin) j= j= Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 14 of 15 and (m –)! (,m) C =–.(.) (πin) Here the convergence is uniform. Next, assume that =for some integers r, m with m ≥ r ≥ . Then γ () = γ (). r,m r,m r,m Hence γ (x)is piecewise C and discontinuous with jump discontinuities at integers. r,m Then the Fourier series of γ (x)converges pointwise to γ (x)for x ∈/ Z and con- r,m r,m verges to   γ () + γ () = γ () + r,m r,m r,m r,m     = B ··· B + (.) c c r,m c ··· c   r c +c +···+c =m,c ,...,c ≥ r r    for x ∈ Z. Now, we can state our second result. Theorem . For all integers s, lwith l ≥ s ≥ , we let s B ··· B c c s,l a c ··· c  a ≤a≤s– c +c +···+c =l+a–s,c ,...,c ≥   a  a s B ··· B c c  a = δ + .(.) s,l a c ··· c  a ≤a≤s– c +c +···+c =l+a–s,c ,...,c ≥   a  a (r,m) (r,m) Assume that = for some integers r, mwithm ≥ r ≥ . Let C , C (n =) be r,m n as in Theorem .. Then we have the following. (r,m) (r,m) πinx C + C e  n n=–∞,n = ⎨  B (x) ··· B (x) for x ∈/ Z, c c c +c +···+c =m,c ,...,c ≥ c ···c    r  r r = (.)   B ··· B + for x ∈ Z. c c r,m c +c +···+c =m,c ,...,c ≥ c ···c     r  r r Acknowledgements The third author is appointed as a chair professor at Tianjin Polytechnic University by Tianjin City in China from August 2015 to August 2019. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the manuscript and typed, read, and approved the final manuscript. Author details 1 2 Department of Mathematics, Texas A&M University - Kingsville, Kingsville, TX 78363, USA. Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea. Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea. Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, 300160, China. Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, Gyeongsangnamdo 52828, Republic of Korea. Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 15 of 15 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 28 February 2017 Accepted: 29 July 2017 References 1. Bayad, A, Kim, T: Higher recurrences for Apostol-Bernoulli-Euler numbers. Russ. J. Math. Phys. 16(1), 1-10 (2012) 2. Ding, D, Yang, J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 20(1), 7-21 (2010) 3. Kim, DS, Kim, T: A note on higher-order Bernoulli polynomials. J. Inequal. Appl. 2013, 111 (2013) 4. Kim, DS, Kim, T, Kim, YH, Lee, S-H: Some arithmetic properties of Bernoulli and Euler numbers. Adv. Stud. Contemp. Math. (Kyungshang) 22(4), 467-480 (2012) 5. Kim, T: Some identities for the Bernoulli, the Euler and Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 20(1), 23-28 (2015) 6. Kim, T: Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 20(1), 23-28 (2010) 7. Liu, H, Wang, W: Some identities on the the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums. Discrete Math. 309, 3346-3363 (2009) 8. Miki, H: A relation between Bernoulli numbers. J. Number Theory 10(3), 297-302 (1978) 9. Srivastava, HM: Some generalizations and basic extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 5(3), 390-414 (2011) 10. Washington, LC: Introduction to Cyclotomic Fields, 2nd edn. Graduate Texts in Mathematics, vol. 83. Springer, New York (1997) 11. Kim, T, Kim, DS, Rim, S-H, Dolgy, D-V: Fourier series of higher-order Bernoulli functions and their applications. J. Inequal. Appl. 2017, 8 (2017) 12. Marsden, JE: Elementary Classical Analysis. Freeman, San Francisco (1974) 13. Zill, DG, Cullen, MR: Advanced Engineering Mathematics. Jones & Bartlett, Boston (2006) 14. Faber, C, Pandharipande, R: Hodge integrals and Gromov-Witten theory. Invent. Math. 139(1), 173-199 (2000) 15. Kim, DS, Kim, T: Bernoulli basis and the product of several Bernoulli polynomials. Int. J. Math. Math. Sci. 2012,Article ID 463659 (2012) 16. Kim, DS, Kim, T: Some identities of higher order Euler polynomials arising from Euler basis. Integral Transforms Spec. Funct. 24(9), 734-738 (2013) 17. Dunne, GV, Schubert, C: Bernoulli number identities from quantum field theory and topological string theory. Commun. Number Theory Phys. 7(2), 225-249 (2013) 18. Gessel, IM: On Miki’s identities for Bernoulli numbers. J. Number Theory 110(1), 75-82 (2005) 19. Shiratani, K, Yokoyama, S: An application of p-adic convolutions. Mem. Fac. Sci., Kyushu Univ., Ser. 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Sums of finite products of Bernoulli functions

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Copyright © 2017 by The Author(s)
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Mathematics; Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations
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1687-1847
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10.1186/s13662-017-1309-9
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Abstract

Kwangwoon University, Seoul, In this paper, we consider three types of functions given by sums of finite products of 139-701, Republic of Korea Department of Mathematics, Bernoulli functions and derive their Fourier series expansions. In addition, we express College of Science, Tianjin each of them in terms of Bernoulli functions. Polytechnic University, Tianjin, 300160, China MSC: 11B68; 42A16 Full list of author information is available at the end of the article Keywords: Fourier series; sums of finite products of Bernoulli functions 1 Introduction As is well known, the Bernoulli polynomials B (x) are given by the generating function t t xt e = B (x) (see [–]). (.) e – m! m= When x =, B = B () are called Bernoulli numbers. For any real number x,welet m m x = x –[x] ∈ [, ) (.) denote the fractional part of x. Fourier series expansion of higher-order Bernoulli functions was treated in the recent paper []. Here we will consider the following three types of functions given by sums of finite products of Bernoulli functions and derive their Fourier series expansions. In addition, we will express each of them in terms of Bernoulli functions. () α (x)= B (x)B (x) ··· B (x) (m ≥ ); m c c c   r c +c +···+c =m,c ,...,c ≥   r  r () β (x) = B (x)B (x) ··· B (x) (m ≥ ); m c c c c +c +···+c =m,c ,...,c ≥   r   r  r c !c !···c !   r () γ (x)= B (x)B (x) ··· B (x) (m ≥ r). r,m c c c c +c +···+c =m,c ,...,c ≥   r   r  r c c ···c   r For elementary facts about Fourier analysis, the reader may refer to any book (for example, see [, ]). As to β (x), we note that the next polynomial identity follows immediately from The- orems . and ., which is in turn derived from the Fourier series expansion of β (x): j–   r B (x)B (x) ··· B (x)=  +  B (x), c c c m+ m–j+ j   c !c ! ··· c ! r j!   r c +c +···+c =m   r j= © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 2 of 15 where r B B ··· B c c c   a = .(.) a c !c ! ··· c !   a max{,r–l}≤a≤r– c +c +···+c =l+a–r   a The obvious polynomial identities can be derived also for α (x)and γ (x)from The- m m orems . and .,and Theorems . and ., respectively. It is remarkable that from the m– Fourier series expansion of the function B (x)B (x)we can derive the k m–k k= k(m–k) Faber-Pandharipande-Zagier identity (see [–]) and the Miki identity (see [–]). 2 The function α (x) Let α (x)= B (x)B (x) ··· B (x)(m ≥ ). Here the sum runs over all non- m c c c c +c +···+c =m   r   r negative integers c , c ,..., c with c + c + ··· + c = m (r ≥ ). Then we will consider the   r   r function α x = B x B x ··· B x,(.) m c c c   c +c +···+c =m   r defined on (–∞, ∞), which is periodic with period . The Fourier series of α (x)is (m) πinx A e , (.) n=–∞ where (m) –πinx A = α x e dx –πinx = α (x)e dx.(.) Before proceeding further, we need to observe the following. α (x)= c B (x)B (x) ··· B (x)  c – c c m r   c +c +···+c =m   r + ··· + c B (x)B (x) ··· B (x)B (x) r c c c c –   r– r = c B (x)B (x) ··· B (x)  c – c c   r c +c +···+c =m,c ≥   r  + ··· + c B (x)B (x) ··· B (x) r c – c c   r c +c +···+c =m,c ≥ r r   =(m + r –) B (x)B (x) ··· B (x) c c c   c +c +···+c =m–   =(m + r –)α (x). (.) m– From this, we have α (x) m+ = α (x)(.) m + r Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 3 of 15 and α (x) dx = α () – α ().(.) m m+ m+ m + r For m ≥ , we put = α () – α () m m m = B ()B () ··· B () – B B ··· B c c c c c c   r   r c +c +···+c =m   = (B + δ ) ··· (B + δ )– B B ··· B c ,c c ,c c c c r r r     c +c +···+c =m   r = B B ··· B – B B ··· B c c c c c c   a   r ≤a≤r c +c +···+c =m+a–r c +c +···+c =m   a   r a≥r–m = B B ··· B,(.) c c c   a max{,r–m}≤a≤r– c +c +···+c =m+a–r   a where we understand that, for r – m ≤ and a = , the inner sum is δ . m,r Observe here that the sum over all c + c + ··· + c = m of any term with a of B and b   r c of δ ( ≤ e, f ≤ r, a + b = r), all give the same sum ,c B ··· B δ ··· δ c c ,c ,c  a a+ a+b c +c +···+c =m   = B B ··· B,(.) c c c   c +c +···+c =m+a–r   a which is not an empty sum as long as m + a – r ≥ , i.e., a ≥ r – m. Thus α () = α () ⇐⇒  = (.) m m m and α (x) dx = .(.) m m+ m + r (m) Now, we are ready to determine the Fourier coefficients A . Case : n =. (m) –πinx A = α (x)e dx   –πinx  –πinx =– α (x)e + α (x)e dx πin πin m + r –  (m–) = A – πin πin m + r – m + r –   (m–) = A –  – m– m πin πin πin πin Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 4 of 15 (m + r –) (m + r –)  j– (m–) = A – m–j+  j (πin) (πin) j= = ··· (m + r –) (m + r –) m j– () = A – m–j+ m j (πin) (πin) j=  (m + r) =– ,(.) m–j+ m + r (πin) j= ()  –πinx where A = e dx =. Case : n =. (m) A = α (x) dx = .(.) m m+ m + r Let us recall the following facts about Bernoulli functions B (x): (a) for m ≥ , πinx B x =–m! ; (.) (πin) n=–∞ n = (b) for m =, πinx ⎨ B (x) for x ∈/ Z, – = (.) πin  for x ∈ Z. n=–∞ n = α (x)(m ≥ ) is piecewise C .Moreover, α (x) is continuous for those positive m m integers m with  =  and discontinuous with jump discontinuities at integers for those positive integers m with  =. Assume first that m is a positive integer with  =.Then α () = α (). Hence α (x) m m m m is piecewise C and continuous. Thus the Fourier series of α (x) converges uniformly to α (x), and α x = m m+ m + r ∞ m (m + r) πinx + –  e m–j+ m + r (πin) n=–∞,n = j= m ∞ πinx   m + r e =  +  –j! m+ m–j+ m + r m + r j (πin) n=–∞ j= n =   m + r =  +  B x m+ m–j+ j m + r m + r j j= B (x)for x ∈/ Z, +  × (.) for x ∈ Z. Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 5 of 15 We can now state our first result. Theorem . For each positive integer l, we let = B B ··· B . l c c c   max{,r–l}≤a≤r– c +c +···+c =l+a–r   a Assume that  = for a positive integer m. Then we have the following. (a) B (x)B (x) ··· B (x) has the Fourier series expansion c c c c +c +···+c =m     r B x B x ··· B x c c c   c +c +···+c =m   r ∞ m (m + r)   πinx =  –  e , m+ m–j+ m + r m + r (πin) n=–∞ j= n = for all x ∈ R, where the convergence is uniform. (b) B x B x ··· B x c c c   r c +c +···+c =m   r   m + r =  +  B x , m+ m–j+ j m + r m + r j j= for all x ∈ R, where B (x) is the Bernoulli function. Assume next that  =  for a positive integer m.Then α () = α (). Hence α (x)is m m m m piecewise C and discontinuous with jump discontinuities at integers. The Fourier series of α (x)converges pointwise to α (x)for x ∈/ Z and converges to m m   α () + α () = α () +  (.) m m m m   for x ∈ Z. Now, we can state our second result. Theorem . For each positive integer l, we let = B B ··· B . l c c c   a max{,r–l}≤a≤r– c +c +···+c =l+a–r   Assume that  = for a positive integer m. Then we have the following. ∞ m   (m + r) πinx (a)  + –  e m+ m–j+ m + r m + r (πin) n=–∞ j= n = B (x)B (x) ··· B (x) for x ∈/ Z, c c c c +c +···+c =m   r   ⎩  B B ··· B +  for x ∈ Z. c c c m c +c +···+c =m   r   r  Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 6 of 15   m + r (b)  +  B x m+ m–j+ j m + r m + r j j= = B x B x ··· B x for x ∈/ Z; c c c   r c +c +···+c =m   r   m + r +  B x m+ m–j+ j m + r m + r j j= = B B ··· B +  for x ∈ Z. c c c m   r c +c +···+c =m   r 3 The function β (x) Let β (x)= B (x)B (x) ··· B (x)(m ≥ ).Herethe sumrunsoverall m c c c c +c +···+c =m   r   r c !c !···c !   r nonnegative integers c , c ,..., c with c + c + ··· + c = m (r ≥ ). Then we will consider   r   r the function β x = B x B x ··· B x,(.) m c c c   r c !c ! ··· c !   r c +c +···+c =m   defined on (–∞, ∞), which is periodic with period . The Fourier series of β (x)is (m) πinx B e,(.) n=–∞ where   (m) –πinx –πinx B = β x e dx = β (x)e dx. (.) m m   Before proceeding further, we need to observe the following. β (x)= B (x)B (x) ··· B (x) c – c c m   r c !c ! ··· c !   r c +c +···+c =m   + ··· + B (x)B (x) ··· B (x) c c c –   c !c ! ··· c !   r = B (x)B (x) ··· B (x) c – c c   r (c –)!c ! ··· c !   r c +c +···+c =m,c ≥   r  + ··· + B (x)B (x) ··· B (x) c c c –   r c !c ! ··· (c –)!   r c +c +···+c =m,c ≥   r r = r B (x)B (x) ··· B (x) c c c   c !c ! ··· c !   r c +c +···+c =m–   r = rβ (x). (.) m– From this, we have β (x) m+ = β (x)(.) r Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 7 of 15 and β (x) dx = β () – β ().(.) m m+ m+ Let = β () – β () m m m B ()B () ··· B () B B ··· B c c c c c c   r   r = – c !c ! ··· c ! c !c ! ··· c !   r   r c +c +···+c =m c +c +···+c =m   r   r (B + δ )(B + δ ) ··· (B + δ ) c ,c c ,c c ,c r r     c !c ! ··· c !   r c +c +···+c =m   r B B ··· B c c c   r c !c ! ··· c !   r c +c +···+c =m   r r B B ··· B c c c   a = ,(.) a c !c ! ··· c !   a c +c +···+c =m+a–r max{,r–m}≤a≤r– a   where we understand that, for r – m ≤ and a = , the inner sum is δ . m,r Observe here that the sum over all c + c + ··· + c = m of any term with a of B and b   r c of δ ( ≤ e, f ≤ r, a + b = r), all give the same sum ,c B ··· B δ ··· δ c c ,c ,c  a+ a+b c !c ! ··· c !   r c +c +···+c =m   r B B ··· B c c c   = ,(.) c !c ! ··· c !   r c +c +···+c =m+a–r   a which is not an empty sum as long as m + a – r ≥ , i.e., a ≥ r – m. Also, we have β () = β () ⇔  = (.) m m m and β (x) dx = .(.) m m+ (m) Now, we would like to determine the Fourier coefficients B . Case : n =. (m) –πinx B = β (x)e dx   –πinx  –πinx =– β (x)e + β (x)e dx πin πin  r –πinx =– β () – β () + β (x)e dx m m m– πin πin  Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 8 of 15 r  (m–) = B – πin πin r r   (m–) = B –  – m– m πin πin πin πin j– r r (m–) = B – m–j+ πin (πin) j= = ··· m m j– r r () = B – m–j+ πin (πin) j= j– =– ,(.) m–j+ (πin) j= ()  –πinx where B = e dx =. Case : n =. (m) B = β (x)= .(.) m m+ β (x)(m ≥ ) is piecewise C .Moreover, β (x) is continuous for those positive m m integers m with  =  and discontinuous with jump discontinuities at integers for those positive integers m with  =. Assume first that  =  for a positive integer m.Then β () = β (). Hence β (x)is m m m m piecewise C and continuous. Thus the Fourier series of β (x) converges uniformly to β (x), and ∞ m j–  r πinx β x =  + –  e m m+ m–j+ r (πin) n=–∞ j= n = m ∞ j– πinx  r e =  +  × –j! m+ m–j+ r j! (πin) n=–∞ j= n = j–  r =  +  B x m+ m–j+ j r j! j= B (x)for x ∈/ Z, +  × (.) for x ∈ Z. Now, we can state our first result. Theorem . For each positive integer l, we let r B B ··· B c c c   = .(.) a c !c ! ··· c !   a max{,r–l}≤a≤r– c +c +···+c =l+a–r   a Assume that  = for a positive integer m. Then we have the following. m Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 9 of 15 (a) B (x)B (x) ··· B (x) has the Fourier series expansion c c c c +c +···+c =m   r r c !c !···c !     r B x B x ··· B x c c c   r c !c ! ··· c !   r c +c +···+c =m   r ∞ m j–  r πinx =  –  e , (.) m+ m–j+ r (πin) n=–∞ j= n = for all x ∈ (–∞, ∞), where the convergence is uniform. (b) B x B x ··· B x c c c   c !c ! ··· c !   r c +c +···+c =m   r j–  r =  +  B x,(.) m+ m–j+ j r j! j= for all x ∈ (–∞, ∞), where B (x) is the Bernoulli function. Assume next that m is a positive integer with  =.Then β () = β (). Hence β (x) m m m m is piecewise C and discontinuous with jump discontinuities at integers. Thus the Fourier series of β (x)converges pointwise to β (x)for x ∈/ Z and converges to m m   β () + β () = β () + m m m m     = B B ··· B +  (.) c c c m   r c !c ! ··· c !    r c +c +···+c =m   r for x ∈ Z. Now, we can state our second result. Theorem . For each positive integer l, let r B B ··· B c c c   = .(.) a c !c ! ··· c !   a max{,r–l}≤a≤r– c +c +···+c =l+a–r   a Assume that  = for a positive integer m. Then we have the following. ∞ m j–  r πinx (a)  –  e m+ m–j+ r (πin) n=–∞,n = j= ⎨  B (x)B (x) ··· B (x) for x ∈/ Z, c c c c +c +···+c =m c !c !···c !     r r     B B ··· B +  for x ∈ Z. c c c m c +c +···+c =m c !c !···c !      r r   Here the convergence is pointwise. j–  r (b)  +  B x m+ m–j+ j r j! j= = B x B x ··· B x for x ∈/ Z, c c c   r c !c ! ··· c !   r c +c +···+c =m   r Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 10 of 15 j–  r +  B x m+ m–j+ j r j! j=   = B B ··· B +  for x ∈ Z. c c c m   r c !c ! ··· c !    r c +c +···+c =m   r Here B (x) is the Bernoulli function. 4 The function γ (x) r,m Let γ (x)= B (x)B (x) ··· B (x)(m ≥ r ≥ ). Here the sum r,m c c c c +c +···+c =m,c ,...,c ≥ c c ···c     r  r r   is over all positive integers c , c ,..., c with c + c + ··· + c = m.   r   r γ (x)= B (x)B (x) ··· B (x) c – c c r,m   r c ··· c  r c +c +···+c =m,c ,...,c ≥   r  r + B (x)B (x) ··· B (x) c c – c   r c c ··· c   r c +c +···+c =m,c ,...,c ≥   r  r + ··· + B (x)B (x) ··· B (x) c c c –   r c c ··· c   r– c +c +···+c =m,c ,...,c ≥   r  r = B (x) ··· B (x) c c  r c ··· c  r c +···+c =m–,c ,...,c ≥  r  r + B (x) ··· B (x) c c  r c ··· c  r c +···+c =m–,c ,...,c ≥  r  r + ··· + B (x) ··· B (x) c c  r– c c ··· c   r– c +c +···+c =m–,c ,...,c ≥   r–  r– + B (x) ··· B (x) c c c c ··· c   r– c +c +···+c =m–,c ,...,c ≥ r r    = rγ (x)+(m –)γ (x). (.) r–,m– r,m– Thus, γ (x)= rγ (x)+(m –)γ (x)(m ≥ r), (.) r–,m– r,m– r,m with γ (x)=. r,r– Replacing m by m +, we get mγ (x)= γ (x)– rγ (x). (.) r,m r–,m r,m+ Denoting γ (x) dx by a ,wehave r,m r,m r  a =– a + , (.) r,m r–,m r,m+ m m where = γ () – γ (). From the recurrence relation (.), we can easily show that r,m r,m r,m r– (r) j– j– γ (x) dx = (–) ,(.) r,m r–j+,m+ j= Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 11 of 15 = γ () – γ () r,m r,m r,m B () ··· B () c c  r c ··· c  r c +c +···+c =m,c ,...,c ≥   r  r B ··· B c c c ··· c  r c +c +···+c =m,c ,...,c ≥   r  r = (B + δ ) ··· (B + δ ) c ,c c ,c   r r c +c +···+c =m,c ,...,c ≥   r  r B ··· B c c c ··· c  r c +c +···+c =m,c ,...,c ≥ r r    r B B ··· B c c c   = . (.) a c c ··· c   a ≤a≤r– c +c +···+c =m+a–r,c ,...,c ≥   a  a Observe here that the sum over all positive integers c ,..., c satisfying c + c + ··· + c =  r   r m of any term with a of B and b of δ ( ≤ e, f ≤ r, a + b = r), all give the same sum c ,c B ··· B δ ··· δ c c ,c ,c  a a+ a+b c c ··· c   r c +c +···+c =m,c ,...,c ≥   r  a B B ··· B c c c   a = ,(.) c c ··· c   a c +c +···+c =m+a–r,c ,...,c ≥   a  a and that, as m + a – r ≥ a,there arenoempty sums. Here we note that, for a = , the inner sum is δ since it corresponds to the sums m,r δ δ ··· δ ,c ,c ,c   r .(.) c c ··· c   r c +c +···+c =m,c ,...,c ≥   r  r Also, γ () = γ () ⇔ =. r,m r,m r,m Now, we would like to consider the function γ x = B x B x ··· B x,(.) r,m c c c   c c ··· c   r c +c +···+c =m,c ,...,c ≥ r r    defined on (–∞, ∞), which is periodic with period . The Fourier series of γ (x)is r,m (r,m) πinx C e,(.) n=–∞ where   (r,m) –πinx –πinx C = γ x e dx = γ (x)e dx.(.) r,m r,m   (r,m) Now, we are going to determine the Fourier coefficients C . n Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 12 of 15 Case : n =. (r,m) –πinx C = γ (x)e dx r,m   –πinx  –πinx =– γ (x)e + γ (x)e dx r,m r,m πin πin   =– γ () – γ () + rγ (x)+(m –)γ (x) r,m r,m r–,m– r,m– πin πin –πinx × e dx   r (r,m–) (r–,m–) =– + C + C .(.) r,m n n πin πin πin From this, we obtain m – r  (r,m) (r,m–) (r–,m–) C = C + C – r,m n n n πin πin πin m – m – r  (r,m–) (r–,m–) = C + C – r,m– n n πin πin πin πin r  (r–,m–) + C – r,m πin πin   (m –) r(m –) (m –) j– j– (r,m–) (r–,m–j) = C + C – r,m–j+ n n  j j (πin) (πin) (πin) j= j= = ··· m–r m–r (m –) r(m –) (m –) m–r j– j– (r,r) (r–,m–j) = C + C – . (.) r,m–j+ n n m–r j j (πin) (πin) (πin) j= j= Here,  r (r,r) –πinx C = x – e dx r  r–   r  –πinx –πinx =– x – e + x – e dx πin  πin  r r    r (r–,r–) =– – – + C,(.) πin   πin and r r   = γ () – γ () = – – . (.) r,r r,r r,r   Thus  r (r,r) (r–,r–) C =– + C.(.) r,r n n πin πin Finally, we obtain, for n =, m–r+ m–r+ r(m –) (m –) j– j– (r,m) (r–,m–j) C = C – .(.) r,m–j+ n n j j (πin) (πin) j= j= Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 13 of 15 Also, we note that, for n =,  (m –)! (,m) –πinx C = B (x)e dx =–.(.) m (πin) (r,m) Thus, for n =, (.)togetherwith(.)determine all C recursively. Case : n =. (r) j– (r,m) j– C = γ (x) dx = (–) .(.) r,m r–j+,m+ j= γ (x)(m ≥ r ≥ ) is piecewise C . In addition, γ (x) is continuous for those pos- r,m r,m itive integers r, m with =  and discontinuous with jump discontinuities at integers r,m for those positive integers r, m with =. r,m Assume first that = for some integers r, m with m ≥ r ≥ . Then γ () = γ (). r,m r,m r,m Hence γ (x)is piecewise C and continuous. Thus the Fourier series of γ (x)con- r,m m verges uniformly to γ (x), and (r,m) (r,m) πinx γ x = C + C e ,  n n=–∞ n = (r,m) (r,m) where C is given by (.), and C ,for each n = , are determined by relations (.) and (.). Now, we are ready to state our first theorem. Theorem . For all integers s, l, with l ≥ s ≥ , we let s B ··· B c c  a s,l a c ··· c  a ≤a≤s– c +···+c =l+a–s,c ,...,c ≥  a  a s B ··· B c c  a = δ + .(.) s,l a c ··· c  a ≤a≤s– c +c +···+c =l+a–s,c ,...,c ≥   a  a Assume that = for some integers r,mwithm ≥ r ≥ . Then we have the following. r,m B (x) ··· B (x) has the Fourier series expansion c c  r c +c +···+c =m,c ,...,c ≥ c ···c   r  r  r B x ··· B x c c c ··· c  r c +c +···+c =m,c ,...,c ≥ r r    (r,m) (r,m) πinx = C + C e ,  n n=–∞,n = (r) (r,m) (,m) (r,m) r– j– j– where C = (–) , with C =, and C , for each n =, are de- j r–j+,m+ n  j=  termined recursively from m–r+ m–r+ r(m –) (m –) j– j– (r,m) (r–,m–j) C = C – ,(.) r,m–j+ n n j j (πin) (πin) j= j= Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 14 of 15 and (m –)! (,m) C =–.(.) (πin) Here the convergence is uniform. Next, assume that =for some integers r, m with m ≥ r ≥ . Then γ () = γ (). r,m r,m r,m Hence γ (x)is piecewise C and discontinuous with jump discontinuities at integers. r,m Then the Fourier series of γ (x)converges pointwise to γ (x)for x ∈/ Z and con- r,m r,m verges to   γ () + γ () = γ () + r,m r,m r,m r,m     = B ··· B + (.) c c r,m c ··· c   r c +c +···+c =m,c ,...,c ≥ r r    for x ∈ Z. Now, we can state our second result. Theorem . For all integers s, lwith l ≥ s ≥ , we let s B ··· B c c s,l a c ··· c  a ≤a≤s– c +c +···+c =l+a–s,c ,...,c ≥   a  a s B ··· B c c  a = δ + .(.) s,l a c ··· c  a ≤a≤s– c +c +···+c =l+a–s,c ,...,c ≥   a  a (r,m) (r,m) Assume that = for some integers r, mwithm ≥ r ≥ . Let C , C (n =) be r,m n as in Theorem .. Then we have the following. (r,m) (r,m) πinx C + C e  n n=–∞,n = ⎨  B (x) ··· B (x) for x ∈/ Z, c c c +c +···+c =m,c ,...,c ≥ c ···c    r  r r = (.)   B ··· B + for x ∈ Z. c c r,m c +c +···+c =m,c ,...,c ≥ c ···c     r  r r Acknowledgements The third author is appointed as a chair professor at Tianjin Polytechnic University by Tianjin City in China from August 2015 to August 2019. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the manuscript and typed, read, and approved the final manuscript. Author details 1 2 Department of Mathematics, Texas A&M University - Kingsville, Kingsville, TX 78363, USA. Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea. Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea. Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, 300160, China. Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, Gyeongsangnamdo 52828, Republic of Korea. Agarwal et al. Advances in Difference Equations (2017) 2017:237 Page 15 of 15 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 28 February 2017 Accepted: 29 July 2017 References 1. Bayad, A, Kim, T: Higher recurrences for Apostol-Bernoulli-Euler numbers. Russ. J. Math. Phys. 16(1), 1-10 (2012) 2. Ding, D, Yang, J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 20(1), 7-21 (2010) 3. Kim, DS, Kim, T: A note on higher-order Bernoulli polynomials. J. Inequal. Appl. 2013, 111 (2013) 4. Kim, DS, Kim, T, Kim, YH, Lee, S-H: Some arithmetic properties of Bernoulli and Euler numbers. Adv. Stud. Contemp. Math. (Kyungshang) 22(4), 467-480 (2012) 5. Kim, T: Some identities for the Bernoulli, the Euler and Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 20(1), 23-28 (2015) 6. Kim, T: Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 20(1), 23-28 (2010) 7. Liu, H, Wang, W: Some identities on the the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums. Discrete Math. 309, 3346-3363 (2009) 8. Miki, H: A relation between Bernoulli numbers. J. Number Theory 10(3), 297-302 (1978) 9. Srivastava, HM: Some generalizations and basic extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 5(3), 390-414 (2011) 10. Washington, LC: Introduction to Cyclotomic Fields, 2nd edn. Graduate Texts in Mathematics, vol. 83. Springer, New York (1997) 11. Kim, T, Kim, DS, Rim, S-H, Dolgy, D-V: Fourier series of higher-order Bernoulli functions and their applications. J. Inequal. Appl. 2017, 8 (2017) 12. Marsden, JE: Elementary Classical Analysis. Freeman, San Francisco (1974) 13. Zill, DG, Cullen, MR: Advanced Engineering Mathematics. Jones & Bartlett, Boston (2006) 14. Faber, C, Pandharipande, R: Hodge integrals and Gromov-Witten theory. Invent. Math. 139(1), 173-199 (2000) 15. Kim, DS, Kim, T: Bernoulli basis and the product of several Bernoulli polynomials. Int. J. Math. Math. Sci. 2012,Article ID 463659 (2012) 16. Kim, DS, Kim, T: Some identities of higher order Euler polynomials arising from Euler basis. Integral Transforms Spec. Funct. 24(9), 734-738 (2013) 17. Dunne, GV, Schubert, C: Bernoulli number identities from quantum field theory and topological string theory. Commun. Number Theory Phys. 7(2), 225-249 (2013) 18. Gessel, IM: On Miki’s identities for Bernoulli numbers. J. Number Theory 110(1), 75-82 (2005) 19. Shiratani, K, Yokoyama, S: An application of p-adic convolutions. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 36(1), 73-83 (1982)

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