Multivariate Goncarov polynomials and integer sequences

Abstract: Univariate delta Gon\v{c}arov polynomials arise when the classical Gon\v{c}arov interpolation problem in numerical analysis is modified by replacing derivatives with delta operators. When the delta operator under consideration is the backward difference operator, we acquire the univariate difference Gon\v{c}arov polynomials, which have a combinatorial relation to lattice paths in the plane with a given right boundary. In this talk, we extend several algebraic and analytic properties of univariate Gon\v{c}arov polynomials to the multivariate case with both the derivative and backward difference operators. We then establish a combinatorial interpretation of multivariate Gon\v{c}arov polynomials in terms of certain constraints on $d$-tuples of integer sequences. This motivates a connection between multivariate Gon\v{c}arov polynomials and a higher-dimensional generalized parking function, the $\mathbf{U}$-parking function, from which we derive several enumerative results based on the theory of delta operators.

This talk is based on joint work with Ayo Adeniran and Lauren Snider.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)