In a previous work we studied generalized Stirling numbers of the second kind Sa1,b1(a2,b2,p2)(p1,k), where a₁, a₂, b₁, b₂ are given complex numbers, a₁, a₂ ̸= 0, and p₁, p₂ are non-negative integers given. In this work we use these generalized Stirling numbers to define Bernoulli polynomials in two variables B_p1,p2 (x₁, x₂), and Euler polynomials in two variables E_p1,p2(x₁, x₂). By using results for S1,x1(1,x2,p2)(p1,k), we obtain generalizations, to the bivariate case, of some well-known properties from the standard case, as addition formulas, difference equations and sums of powers. We obtain some identities for bivariate Bernoulli and Euler polynomials, and some generalizations, to the bivariate case, of several known identities for Bernoulli and Euler numbers and polynomials of the standard case.

Keywords: Bernoulli number, Euler number, Bernoulli polynomial, Euler polynomial, Generalized Stirling number