Hypergeometric degenerate Bernoulli polynomials and numbers
Keywords:Bernoulli numbers, hypergeometric Bernoulli numbers, hypergeometric Cauchy numbers, hypergeometric functions, degenerate Bernoulli numbers, determinants, recurrence relations
Carlitz defined the degenerate Bernoulli polynomials βn(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 βN, n(λ, x) and numbers, in particular, in terms of determinants.
The coefficients of the polynomial βn(λ, 0) were completely determined by Howard in 1996. We determine the coefficients of the polynomial βN, n(λ, 0). Hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers appear in the coefficients.