Multivariate Delta Gon£arov and Abel Polynomials

Classical Gon£arov polynomials are polynomials which interpolate derivatives. Delta Gon£arov polynomials are polynomials which interpolate delta operators, e.g., forward and backward difference operators. We extend fundamental aspects of the theory of classical bivariate Gon£arov polynomials and univariate delta Gon£arov polynomials to the multivariate setting using umbral calculus. After introducing systems of delta operators, we de ne multivariate delta Gon£arov polynomials, show that the associated interpolation problem is always solvable, and derive a generating function (an Appell relation) for them. We show that systems of delta Gon£arov polynomials on an interpolation grid Z ⊆ R are of binomial type if and only if Z = AN for some d×d matrix A. This motivates our de nition of delta Abel polynomials to be exactly those delta Gon£arov polynomials which are based on such a grid. Finally, compact formulas for delta Abel polynomials in all dimensions are given for separable systems of delta operators. This recovers a former result for classical bivariate Abel polynomials and extends previous partial results for classical trivariate Abel polynomials to all dimensions.