Bivariate affine Gončarov polynomials

Bivariate Gončarov polynomials are a basis of the solutions of the bivariate Gončarov Interpolation Problem in numerical analysis. A sequence of bivariate Gončarov polynomials is determined by a set of nodes Z={(xi,j,yi,j)∈R2}Z={(xi,j,yi,j)∈R2} and is an affine sequence if ZZ is an affine transformation of the lattice grid N2N2, i.e.,  (xi,j,yi,j)T=A(i,j)T+(c1,c2)T(xi,j,yi,j)T=A(i,j)T+(c1,c2)T for some 2×2 matrix AA and constants c1,c2c1,c2. In this paper we prove that a sequence of bivariate Gončarov polynomials is of binomial type if and only if it is an affine sequence with c1=c2=0c1=c2=0. Such polynomials form a higher-dimensional analog of the Abel polynomial An(x;a)=x(x−an)n−1An(x;a)=x(x−an)n−1. We present explicit formulas for a general sequence of bivariate affine Gončarov polynomials and its exponential generating function, and use the algebraic properties of Gončarov polynomials to give some new two-dimensional generalizations of Abel identities.