Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems

TL;DR

This paper derives asymptotic expansions for four families of orthogonal polynomials linked to indeterminate moment problems, using difference equation techniques, and proposes a conjecture on their large degree behavior.

Contribution

It provides Plancherel-Rotach asymptotics for specific indeterminate moment problem polynomials and introduces a conjecture on their asymptotic behavior.

Findings

Asymptotic formulas for Chen–Ismail, Berg–Letessier–Valent, Conrad–Flajolet I and II polynomials.

Identification of asymptotic behavior related to indeterminate moment problems.

Proposal of a conjecture on large degree polynomial behavior.

Abstract

We study the Plancherel--Rotach asymptotics of four families of orthogonal polynomials, the Chen--Ismail polynomials, the Berg-Letessier-Valent polynomials, the Conrad--Flajolet polynomials I and II. All these polynomials arise in indeterminate moment problems and three of them are birth and death process polynomials with cubic or quartic rates. We employ a difference equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to a conjecture about large degree behavior of polynomials orthogonal with respect to solutions of indeterminate moment problems.