Multivariate Delta Goncarov and Abel Polynomials

TL;DR

This paper extends the theory of Goncarov and Abel polynomials to multivariate settings using umbral calculus, providing new formulas, properties, and characterizations of these polynomials in multiple dimensions.

Contribution

It introduces multivariate delta Goncarov and Abel polynomials, establishes their properties, and characterizes when they are of binomial type, extending classical univariate results to higher dimensions.

Findings

Multivariate delta Goncarov polynomials are always solvable.

They are of binomial type if and only if the grid is of the form Aℕ^d.

Explicit formulas for delta Abel polynomials in all dimensions are provided.

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.