Identities of symmetry for q-Euler polynomials

TL;DR

This paper introduces new three-variable symmetry identities for q-Euler polynomials and q-analogue alternating power sums, expanding the understanding of their symmetrical properties and deriving novel corollaries.

Contribution

It presents the first derivation of three-variable symmetry identities for q-Euler polynomials and q-analogue sums, extending previous two-variable results.

Findings

Eight basic symmetry identities in three variables

New corollaries from the symmetry identities

Enhanced understanding of q-Euler polynomial symmetries

Abstract

In this paper, we derive eight basic identities of symmetry in three variables related to $q$-Euler polynomials and the $q$-analogue of alternating power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundance of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on the $p$-adic integral expression of the generating function for the $q$-Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the $q$-analogue of alternating power sums.