Bernoulli and Euler Polynomials in Two Variables

In a previous work we studied generalized Stirling numbers of the second kind S(a2,b2,p2)a1,b1(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 Bp1,p2 (x1, x2), and Euler polynomials in two variables Ep1,p2(x1, x2). By using results for S(1,x2,p2)1,x1(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. ©Kyungpook Mathematical Journal