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Several properties of hypergeometric Bernoulli numbers. (English) Zbl 1498.11071

Summary: In this paper, we give several characteristics of hypergeometric Bernoulli numbers, including several identities for hypergeometric Bernoulli numbers which the convergents of the continued fraction expansion of the generating function of the hypergeometric Bernoulli numbers entail. We show an analog of Kummer’s congruences in the classical Bernoulli numbers. We also give some determinant expressions of hypergeometric Bernoulli numbers and some relations between the hypergeometric and the classical Bernoulli numbers. By applying Trudi’s formula, we have some different expressions and inversion relations.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11A55 Continued fractions
11B37 Recurrences
11C20 Matrices, determinants in number theory
15A15 Determinants, permanents, traces, other special matrix functions
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
05A15 Exact enumeration problems, generating functions
05A19 Combinatorial identities, bijective combinatorics
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References:

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