Generalized Rook Polynomials and Orthogonal Polynomials

Abstract

We consider several generalizations of rook polynomials. In particular we develop analogs of the theory of rook polynomials that are related to general Laguerre and Charlier polynomials in the same way that ordinary rook polynomials are related to simple Laguerre polynomials.

A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials

“…The proof of Theorem 1.1 is of combinatorial nature, this is in the same vein as [8,10,13], where the authors were looking for a unified and more transparent approach to the linearization problems. More precisely, we first set up a combinatorial model for the Sheffer polynomials, combining with the combinatorial interpretation for the moment L in (2) we then interpret the linearization coefficients as the generating functions of some finite structures.…”

“…Finally, we derive from (8) and (10) a new interpretation for the linearization coefficient of Meixner-Pollaczek polynomials. …”

“…The problem of finding interesting formulas for the linearization coefficients of the classical polynomials have attracted much attention in the last three decades (see [2,5,7,8,13,15,17] and the references cited there). Once the moment sequence L(x n ) is determined, this problem can be considered as a formal algebraic calculus.…”

From a Zoo to a Zoology: Towards a General Theory of Graph Polynomials

“…Chung and R.L. Graham [19], and independently in the context of rook polynomials, by I. Gessel, [37]. In [19] it is presented as an attempt to create a Tutte-like polynomial for directed graphs, and is closely related to the chromatic polynomial.…”

Multilinear generating functions for Charlier polynomials

“…Charlier polynomials have been studied using combinatorial methods in [5], [8], [10], [11], [15], and [16]. In this paper, we prove a multilinear generating function for Charlier polynomials using the combinatorial model of Charlier configurations [10,11] and the approach of Foata and Garsia [6] in their proof of Slepian's multilinear extension of the Mehler formula for Hermite polynomials [12].…”