Generalized Goncarov polynomials
Rudolph Lorentz, Salvatore Tringali, and Catherine H. Yan

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 of polynomials defined by the biorthogonality relation for all , where is a delta operator, a sequence of scalars, and the evaluation at . 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.
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