Several properties of hypergeometric Bernoulli numbers

Abstract

In this paper, we give the determinant expressions of the hypergeometric Bernoulli numbers, and some relations between the hypergeometric and the classical Bernoulli numbers which include Kummer's congruences. By applying Trudi's formula, we have some different expressions and inversion relations. We also determine explicit forms of convergents of the generating function of the hypergeometric Bernoulli numbers, from which several identities for hypergeometric Bernoulli numbers are given.

A new class of higher order hypergeometric Bernoulli polynomials associated with Hermite polynomials

“…Corollary 2.4. For x = y = 0 in (2.7), the result reduces to the known result of Aoki et al [2] as follows…”

“…For N, r ∈ N, the higher-order hypergeometric Bernoulli polynomials B (r) N,n (x) are defined by means of the generating function, (see [2], [7], [10])…”

Two types of hypergeometric degenerate Cauchy numbers

“…which is equal to the expression of B N,n λ n in[15, Proposition 2]. From Theorem 3 again, we get the constant term of γ N,n (λ) as…”

“…We list some initial values of γ N,n (λ) (0 ≤ n ≤ 5) in Appendix. When λ → 0 in (15), c N,n = γ N,n (0) are the hypergeometric Cauchy numbers in (14). When N = 1 in (15), γ n (λ) = γ 1,n (λ) are the degenerate Cauchy numbers in (9).…”

Hypergeometric Euler numbers

“…When N = 0, then E n = E 0,n are classical Euler numbers defined in (1). In [18], the truncated Euler polynomial E m,n (x) is introduced as a generalization of the classical Euler polynomial E n (x).…”