Multivariate Difference Gon\v{c}arov Polynomials

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 paper, we extend several algebraic and analytic properties of univariate difference Gon\v{c}arov polynomials to the multivariate case. We then establish a combinatorial interpretation of multivariate difference Gon\v{c}arov polynomials in terms of certain constraints on $d$-tuples of non-decreasing integer sequences. This motivates a connection between multivariate difference Gon\v{c}arov polynomials and a higher-dimensional generalized parking function, the $\boldsymbol{U}$-parking function, from which we derive several enumerative results based on the theory of multivariate delta Gon\v{c}arov polynomials.