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 +  λ t ) 1/ λ  − 1) −1 (1 +  λ t ) 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.