Some formulas for fully degenerate Bernoulli numbers and polynomials

TL;DR

This paper introduces and explores fully degenerate Bernoulli numbers and polynomials, providing explicit formulas and representations in terms of degenerate Stirling numbers and polynomials, extending classical Bernoulli concepts.

Contribution

It presents new explicit formulas for fully degenerate Bernoulli numbers and polynomials using degenerate Stirling numbers, enriching the theory of degenerate special functions.

Findings

Derived explicit expressions for degenerate Bernoulli polynomials and numbers.

Connected degenerate Bernoulli polynomials to degenerate Stirling numbers.

Provided representations for degenerate poly-Bernoulli polynomials.

Abstract

The aim of this paper is to study the fully degenerate Bernoulli polynomials and numbers, which are a degenerate version of Bernoulli polynomials and numbers and arise naturally from the Volkenborn integral of the degenerate exponential functions on Zp. We find some explicit expressions for the fully degenerate Bernoulli polynomials and numbers in terms of the degenerate Stirling numbers of the second kind, the degenerate r-Stirling numbers of the second kind and of the degenerate Stirling polynomials. We also consider the degenerate poly-Bernoulli polynomials and derive explicit representations for them in terms of the same degenerate Stirling numbers and polynomials.