Computation of Al-Salam Carlitz and Askey-Wilson Moments using Motzkin Paths

Gaspard Ohlmann

doi:10.37236/9780

Abstract

In this paper we study the moments of polynomials from the Askey scheme, and we focus on Askey-Wilson polynomials. More precisely, we give a combinatorial proof for the case where $d=0$. Their values have already been computed by Kim and Stanton in 2015, however, the proof is not completely combinatorial, which means that an explicit bijection has not been exhibited yet. In this work, we use a new combinatorial approach for the simpler case of Al-Salam-Carlitz, using a sign reversing involution that directly operates on Motzkin path. We then generalize this method to Askey-Wilson polynomials with $d=0$ only, providing the first fully combinatorial proof for that case.

Highlights

Enumeration of objects is a powerful tool for understanding their properties and revealing hidden connections between them
Looking at the effects of the sign reversing involution at the Motzkin paths level, we find patterns and behaviors that seem to reproduce in our more general case
We introduce Motzkin paths, which are the objects used in the first combinatorial approach described below

Summary

Introduction

Enumeration of objects is a powerful tool for understanding their properties and revealing hidden connections between them. Stanton and Viennot [5] used q-Hermite Polynomials and inhomogenous matchings and Kim and Stanton [10] recently computed the general moment of Askey Wislon polynomials We compute Askey-Wilson moments using uniquely Motzkin paths, providing a combinatorial proof; using a sign reversing involution on Motzkin paths. In order to find a solution to the previously described problem, we focused on a simpler family of orthogonal polynomials: the Al-Salam-Carlitz polynomials Their moments have already been computed with a combinatorial proof using Striped Skew Shapes by Dongsu Kim. the Striped Skew Shapes seem to be specific to this case and it was not possible to generalize them to our situation.

Objectives

Methods

Conclusion