### 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.

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MANUSCRIPTISSN: 1855-3974

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