A note on multi-Euler-Genocchi and degenerate multi-Euler-Genocchi polynomials

TL;DR

This paper introduces and studies multi-Euler-Genocchi and degenerate multi-Euler-Genocchi polynomials, expressing them in terms of known polynomials and Stirling numbers, expanding the theoretical framework of these special functions.

Contribution

It defines multi-Euler-Genocchi polynomials using multiple logarithms and relates them to existing polynomials and Stirling numbers, providing new formulas and connections.

Findings

Expressed multi-Euler-Genocchi polynomials via generalized Euler-Genocchi polynomials and Stirling numbers.

Expressed degenerate multi-Euler-Genocchi polynomials via generalized degenerate Euler-Genocchi polynomials and Stirling numbers.

Established new relationships between these polynomials and combinatorial numbers.

Abstract

Recently, introduced are the generalized Euler-Genocchi and generalized degenerate Euler-Genocchi polynomials. The aim of this note is to study the multi-Euler-Genocchi and degenerate multi-Euler-Genocchi polynomials which are defined by means of the multiple logarithm and generalize respectively the generalized Euler-Genocchi and generalized degenerate Euler-Genocchi polynomials. Especially, we express the former by the generalized Euler-Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind, and the latter by the generalized degenerate Euler-Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind.