Identities of symmetry for Bernoulli polynomials arising from quotients of Volkenborn integrals invariant under S_3

TL;DR

This paper introduces new identities of symmetry involving Bernoulli polynomials in three variables, derived from $p$-adic integrals, expanding the understanding of symmetric properties beyond two-variable cases.

Contribution

It presents eight new identities of symmetry in three variables for Bernoulli polynomials, derived from $p$-adic integral techniques, which are novel compared to previous two-variable results.

Findings

Eight new symmetry identities in three variables for Bernoulli polynomials

Derivation based on $p$-adic integral expressions of generating functions

Enhanced understanding of symmetry properties in Bernoulli polynomials

Abstract

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