Multivariate delta Gončarov and Abel polynomials

Abstract

Classical Gončarov polynomials are polynomials which interpolate derivatives. Delta Gončarov polynomials are polynomials which interpolate delta operators, e.g., forward and backward difference operators. We extend fundamental aspects of the theory of classical bivariate Gončarov polynomials and univariate delta Gončarov polynomials to the multivariate setting using umbral calculus. After introducing systems of delta operators, we define multivariate delta Gončarov 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 Gončarov polynomials on an interpolation grid Z ⊆ R d are of binomial type if and only if Z = AN d for some d × d matrix A. This motivates our definition of delta Abel polynomials to be exactly those delta Gončarov 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.

Multivariate difference Gončarov polynomials

“…Next we discuss the algebraic properties of bivariate difference Gončarov polynomials, analogous to those of the univariate case. We remark that Theorems 2, 3, and 9 are special cases of Propositions 3.5, 3.6 and Theorem 5.1 in [12], and the other results are new.…”

“…, x d ] that naturally extend the univariate difference Gončarov polynomials to multiple variables. A general theory of systems of delta operators and delta Gončarov polynomials in multi-variables was introduced in [12] . In this paper we only need a special case: the system of delta operators is…”

“…, where ∆ xi is the backward difference operator with respect to the variable x i . We will first state the definition and the basic properties from the general theory established in [12]. Then we present some special algebraic properties of multivariate difference Gončarov polynomials.…”

“…From the general theory developed in [12], we have that for any values {b i,j : (i, j) ∈ S m,n }, the bivariate Gončarov interpolation problem with difference operators has a unique solution in the space Π 2 m,n . In particular, by taking all but one of {b i,j : (i, j) ∈ S m,n } to be 0, we can define the bivariate difference Gončarov polynomials.…”

“…Another profound generalization of Gončarov polynomials is obtained by applying the rich theory of delta operators and finite operator calculus, which is a unified theory on linear operators analogous to the differentiation operator D and special polynomials developed by Rota, Kahaner, and Odlyzko [16]. Replacing D with an arbitrary delta operator in the Gončarov interpolation problem, Lorentz, Tringali, and Yan [11,12] introduced the delta Gončarov polynomials and extended many of the algebraic properties of (classical) Gončarov polynomials to this generalized case. They also studied multivariate delta Gončarov polynomials and characterized those that are of binomial type.…”