On bi-variate poly-Bernoulli polynomials

TL;DR

This paper introduces bi-variate poly-Bernoulli polynomials using generalized Stirling numbers, extending classical Bernoulli polynomial results and deriving new identities and recurrences.

Contribution

It presents the first definition and properties of bi-variate poly-Bernoulli polynomials, generalizing known Bernoulli polynomial results and establishing new identities and recurrence relations.

Findings

Derived addition and binomial formulas for bi-variate poly-Bernoulli polynomials.

Established identities connecting poly-Bernoulli and Bernoulli polynomials.

Proved generalized recurrence relations for the polynomials.

Abstract

We introduce poly-Bernoulli polynomials in two variables by using a generalization of Stirling numbers of the second kind that we studied in a previous work. We prove the bi-variate poly-Bernoulli polynomial version of some known results on standard Bernoulli polynomials, as the addition formula and the binomial formula. We also prove a result that allows us to obtain poly-Bernoulli polynomial identities from polynomial identities, and we use this result to obtain several identities involving products of poly-Bernoulli and/or standard Bernoulli polynomials. We prove two generalized recurrences for bi-variate poly-Bernoulli polynomials, and obtain some corollaries from them.