Hypergeometric degenerate Bernoulli polynomials and numbers

Abstract

Carlitz defined the degenerate Bernoulli polynomials β_n(λ, x) by means of the generating function t((1 + λt)^1/λ − 1)⁻¹(1 + λt)^x/λ. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In this paper, we show some expressions and properties of hypergeometric degenerate Bernoulli polynomials β_N, n(λ, x) and numbers, in particular, in terms of determinants.

The coefficients of the polynomial β_n(λ, 0) were completely determined by Howard in 1996. We determine the coefficients of the polynomial β_N, n(λ, 0). Hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers appear in the coefficients.