  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.
