Some identities related to degenerate Bernoulli and degenerate Euler polynomials

Abstract

The aim of this paper is to study degenerate Bernoulli and degenerate Euler polynomials and numbers and their higher-order analogues. We express the degenerate Euler polynomials in terms of the degenerate Bernoulli polynomials and vice versa. We prove the distribution formulas for degenerate Bernoulli and degenerate Euler polynomials. We obtain some identities among the higher-order degenerate Bernoulli and higher-order degenerate Euler polynomials. We express the higher-order degenerate Bernoulli polynomials in $x + y$x+y as a linear combination of the degenerate Euler polynomials in $y$y. We get certain identities involving the degenerate $r$r-Stirling numbers of the second and the binomial coefficients.