Degenerate Eulerian polynomials and numbers

TL;DR

This paper introduces degenerate Eulerian polynomials and numbers, exploring their properties, identities, and relations to other degenerate combinatorial numbers, expanding the classical Eulerian framework.

Contribution

It provides the first systematic study of degenerate Eulerian polynomials and numbers, including identities, recursive relations, and connections to other degenerate special numbers.

Findings

Derived identities and recursive relations for degenerate Eulerian polynomials and numbers.

Established generating functions and a degenerate Worpitzky's identity.

Connected degenerate Eulerian numbers with degenerate Stirling and Bernoulli numbers.

Abstract

The aim of this paper is to study degenerate Eulerian polynomials and degenerate Eulerian numbers, respectively as degenerate versions of the Eulerian polynomials and the Eulerian numbers, and to derive some of their properties. Specifically, we derive an identity, recursive relations, generating function and degenerate version of Worpitzky's identity for the degenerate Eulerian polynomials and numbers. In addition, we obtain several results involving the degenerate Stirling numbers of the second kind and the degenerate Bernoulli numbers as well as the degenerate Eulerian numbers.