On bi-variate poly-Bernoulli polynomials (Q6585115)

In this paper, poly-Bernoulli polynomials in two variables, called bi-variate poly-Bernoulli polynomials, are studied. The bi-variate poly-Bernoulli polynomials are defined by using a generalization of Stirling numbers of the second kind, which was studied in [\textit{C. Pita-Ruiz}, ``Generalized Stirling numbers. I'', Preprint, \url{arXiv:1803.05953}]. Using the results there, results for the bi-variate poly-Bernoulli polynomials are obtained. It also proves a result that derives the identity of poly-Bernoulli polynomials from the identity of polynomials. By applying this result, several identities involving products of poly-Bernoulli and/or standard Bernoulli polynomials are obtained. Finally, two generalized recurrences for bi-variate poly-Bernoulli polynomials are shown, and some consequences of them are obtained.