Generalized Goncarov polynomials

TL;DR

This paper introduces generalized Goncarov polynomials, a new basis for solving Goncarov interpolation problems with delta operators, and explores their algebraic, analytic, and combinatorial applications.

Contribution

It defines and characterizes generalized Goncarov polynomials, extending classical Goncarov polynomials, and demonstrates their utility in combinatorial enumeration under linear constraints.

Findings

Defined generalized Goncarov polynomials via biorthogonality relations.

Established algebraic and analytic properties of these polynomials.

Applied the polynomials to enumerate combinatorial structures with linear order constraints.

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.