Moments of Orthogonal Polynomials and Exponential Generating Functions

TL;DR

This paper explores the algebraic derivation of moments for classical orthogonal polynomials, examines their exponential generating functions, and connects these moments to specific polynomial families like Dumont-Foata and Askey-Wilson.

Contribution

It introduces an algebraic approach to orthogonal polynomial moments and links these moments to well-known polynomial families with explicit generating functions.

Findings

Derived orthogonality algebraically from moment sequences.

Identified moments of classical orthogonal polynomials with exponential generating functions.

Connected Dumont-Foata and Askey-Wilson polynomials through their moments.

Abstract

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ($q=1$) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont-Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally we show that one of our methods can be applied to deal with the moments of Askey-Wilson polynomials.