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Multivariate Delta Goncarov and Abel Polynomials

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 $Z \subseteq \mathbb{R}^d$ 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 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.