Identities of symmetry for Euler polynomials arising from quotients of fermionic integrals invariant under S_3

TL;DR

This paper introduces new three-variable symmetry identities for Euler polynomials and alternating power sums, expanding the understanding of symmetries beyond the previously studied two-variable cases using p-adic integrals.

Contribution

It derives eight novel three-variable symmetry identities for Euler polynomials, revealing deeper symmetries and their implications.

Findings

Eight new symmetry identities in three variables

Enhanced understanding of Euler polynomial symmetries

Connections between p-adic integrals and generating functions

Abstract

In this paper, we derive eight basic identities of symmetry in three variables related to Euler polynomials and 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 Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating power sums.