Multivariate Delta Goncarov and Abel Polynomials

Abstract

Classical Gon\v{c}arov polynomials are polynomials which interpolate derivatives. Delta Gon\v{c}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\v{c}arov polynomials and univariate delta Gon\v{c}arov polynomials to the multivariate setting using umbral calculus. After introducing systems of delta operators, we define multivariate delta Gon\v{c}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\v{c}arov polynomials on an interpolation grid $Z \subseteq \mathbb{R}^d$ are of binomial type if and only if $Z = A\mathbb{N}^d$ for some $d\times d$ matrix $A$. This motivates our definition of delta Abel polynomials to be exactly those delta Gon\v{c}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.