On a duality formula for certain sums of values of poly-Bernoulli polynomials and its application

TL;DR

This paper establishes a duality formula for sums of poly-Bernoulli polynomial values, generalizing known dualities for poly-Bernoulli numbers, and connects these to Genocchi numbers through analytic methods.

Contribution

It introduces a new duality formula for poly-Bernoulli polynomial sums and provides an analytic proof using zeta-functions, extending previous results.

Findings

Derived generating functions for the sums of poly-Bernoulli polynomial values

Proved the duality formula analytically via zeta-functions

Connected poly-Bernoulli numbers with Genocchi numbers

Abstract

We prove a duality formula for certain sums of values of poly-Bernoulli polynomials which generalizes dualities for poly-Bernoulli numbers. We first compute two types of generating functions for these sums, from which the duality formula is apparent. Secondly we give an analytic proof of the duality from the viewpoint of our previous study of zeta-functions of Arakawa-Kaneko type. As an application, we give a formula that relates poly-Bernoulli numbers to the Genocchi numbers.