Multivariate Difference Goncarov Polynomials
Abstract
Univariate delta Goncarov polynomials arise when the classical Goncarov 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 Goncarov polynomials, which have a combinatorial relation to lattice paths in the plane with a given right boundary. In this paper, we extend several algebraic and analytic properties of univariate difference Goncarov polynomials to the multivariate case. We then establish a combinatorial interpretation of multivariate difference Goncarov polynomials in terms of certain constraints on d-tuples of non-decreasing integer sequences. This motivates a connection between multivariate difference Goncarov polynomials and a higher-dimensional generalized parking function, the U-parking function, from which we derive several enumerative results based on the theory of multivariate delta Goncarov polynomials.