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Bernoulli numbers and confluent hypergeometric functions. (English) Zbl 1140.11309
Bennett, M.A. (ed.) et al., Number theory for the millennium I. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21–26, 2000. Natick, MA: A K Peters (ISBN 1-56881-126-8/hbk). 343-363 (2002).

From the text: If the reciprocal of the confluent hypergeometric function M(s+1;r+s+2;z) is taken as an exponential generating function, the resulting sequences of numbers have many properties resembling those of the Bernoulli numbers, which arise when r=s=0.

Other special cases include van der Pol numbers and generalized van der Pol numbers (for positive integers r=s), as well as various sequences of combinatorial numbers and linear recurrence sequences of arbitrary orders. Explicit expressions resembling Euler’s formula involve sums of powers of the zeros of the corresponding confluent hypergeometric functions.


MSC:
11B68Bernoulli and Euler numbers and polynomials
33C10Bessel and Airy functions, cylinder functions, 0 F 1