Orthogonal Polynomials of Askey-Wilson Type

TL;DR

This paper investigates two families of orthogonal polynomials related to Askey-Wilson polynomials, focusing on their orthogonality properties, measures, and asymptotic behavior, including a finite family on the real line and an infinite indeterminate case.

Contribution

It introduces a finite family of orthogonal polynomials related to Askey-Wilson polynomials with real-line orthogonality and analyzes an infinite family with indeterminate moment problem, providing multiple measures and asymptotics.

Findings

Finite family orthogonal on the real line.

Infinite family with indeterminate moment problem.

Derived Plancherel-Rotach asymptotics.

Abstract

We study two families of orthogonal polynomials. The first is a finite family related to the Askey-Wilson polynomials but the orthogonality is on the real line. A limiting case of this family is an infinite system of orthogonal polynomials whose moment problem is indeterminate. We provide several orthogonality measures for the infinite family and derive their Plancherel-Rotach asymptotics.