Properties of the van der Pol numbers and polynomials.

F.T. Howard

doi:10.1515/crll.1973.260.35

$By F. T. Howard in Winston-Salem 1. Introduction In [12], p. 235, B. van der Pol defines the sequence of rational numbers F 0 , F1? F 2 , . . . by means of (l 1) ( J _x __ __ y· F __ 6x(e + 1) -- 12(e* -- 1) ~~ ± n n\ ' /y.3 TMW The first few values of FM are F0 = l, Fa = -- 1/2, F 2 = 1/5, F3 = -- 1/20. These numbers, which we call the van der Pol numbers, are used in the process of "unsmoothing" a "smoothed" function of three variables. As pointed out in [7], they are also of interest because of their relationship to the zeros of the Bessel function. If Jv(z) denotes the Bessel function of the first kind and (1. 2) mr2n m=l (n = l, 2, . . .) where the jVtm are the zeros of z~~v Jv(z), then (1. 3) er2,,(3/2) = (-ir'S · 22-1F2B/(2re)! (n > l). The van der Pol numbers are also related to the Bernoulli numbers Bn defined by (L4) e-^r = ,, ^' rpft (L 5) exax2 6s(e* + l) -- 12(e* -- 1)$

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Properties of the van der Pol numbers and polynomials.

Howard , F.T.

Follow Journal für die reine und angewandte Mathematik (Crelle's Journal) , Volume 1973 (260) de Gruyter – Jan 1, 1973

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Properties of the van der Pol numbers and polynomials.

Howard , F.T.

Follow Journal für die reine und angewandte Mathematik (Crelle's Journal) , Volume 1973 (260) de Gruyter – Jan 1, 1973

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