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} and is an affine sequence if Z is an affine transformation of the lattice grid N2, i.e., (xi,j, yi,j) = A(i, j)T + (c1, c2) for some 2 × 2 matrix A and constants c1, c2. In this paper we prove that a sequence of bivariateGončarovpolynomials is of binomial type if andonly if it is an affine sequencewith c1 = c2 = 0. Such polynomials form a higher-dimensional analog of the Abel polynomial An(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. © 2016 Elsevier B.V. All rights reserved.