Carlitz defined the degenerate Bernoulli polynomials ?$\beta_n(\lambda,x)$? by means of the generating function ?$t\left((1+\lambda t)^{1/\lambda}-1\right)^{-1} (1+\lambda t)^{x/\lambda})$?. 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 ?$\beta_{N,n}(\lambda, x)$? and numbers, in particular, in terms of determinants. The coefficients of the polynomial ?$\beta_n(\lambda, 0)$? were completely determined by Howard in 1996. We determine the coefficients of the polynomial ?$\beta_{N,n}(\lambda, 0)$?. Hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers appear in the coefficients.

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URN

URN:NBN:SI:DOC-JSUIUD70

COBISSID

39896579

Added

28.04.2021

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Citation

APA:

Komatsu, Takao (2020). Hypergeometric degenerate Bernoulli polynomials and numbers. Ars mathematica contemporanea, volume 18, issue 1, str. 163-177. URN:NBN:SI:DOC-JSUIUD70 from http://www.dlib.si

MLA:

Komatsu, Takao. "Hypergeometric degenerate Bernoulli polynomials and numbers." Ars mathematica contemporanea volume 18. issue 1 (2020) str. 163-177.
<http://www.dlib.si/?URN=URN:NBN:SI:DOC-JSUIUD70>

Ars mathematica contemporanea
2008-
(Assembly record)