Generalized Goncarov polynomials

Abstract

We introduce the sequence of generalized Gon\v{c}arov polynomials, which is a basis for the solutions to the Gon\v{c}arov interpolation problem with respect to a delta operator. Explicitly, a generalized Gon\v{c}arov basis is a sequence $(t_n(x))_{n \ge 0}$ of polynomials defined by the biorthogonality relation $\varepsilon_{z_i}(\mathfrak d^{i}(t_n(x))) = n! \;\! \delta_{i,n}$ for all $i,n \in \mathbf N$, where $\mathfrak d$ is a delta operator, $\mathcal Z = (z_i)_{i \ge 0}$ a sequence of scalars, and $\varepsilon_{z_i}$ the evaluation at $z_i$. We present algebraic and analytic properties of generalized Gon\v{c}arov polynomials and show that such polynomial sequences provide a natural algebraic tool for enumerating combinatorial structures with a linear constraint on their order statistics.