Identities of symmetry for generalized twisted Bernoulli polynomials twisted by unramified roots of unity

TL;DR

This paper establishes new symmetry identities involving generalized twisted Bernoulli polynomials and power sums twisted by unramified roots of unity, expanding previous work on ramified roots.

Contribution

It derives multiple symmetry identities in two and three variables for these polynomials and sums, using $p$-adic integral methods.

Findings

Three identities of symmetry in two variables

Eight identities of symmetry in three variables

Extension of previous results to unramified roots of unity

Abstract

We derive three identities of symmetry in two variables and eight in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by unramified roots of unity. The case of ramified roots of unity was treated previously. The derivations of identities are based on the $p$-adic integral expression, with respect to a measure introduced by Koblitz, of the generating function for the generalized twisted Bernoulli polynomials and the quotient of $p$-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.