Plancherel_Rotach Asymptotics for q-Orthogonal Polynomials

TL;DR

This paper derives Plancherel-Rotach asymptotics near the largest zero for certain q-orthogonal polynomials, including q-Hermite, q-Laguerre, and Stieltjes-Wigert, expanding understanding of their asymptotic behavior.

Contribution

It provides new asymptotic formulas at the soft edge for classes of q-orthogonal polynomials and introduces a one-parameter family of solutions to the Ramanujan q-difference equation.

Findings

Asymptotics established for q-Hermite, q-Laguerre, and Stieltjes-Wigert polynomials.

New solutions to the Ramanujan q-difference equation introduced.

Enhanced understanding of the behavior of q-orthogonal polynomials near their largest zeros.

Abstract

We establish the Plancherel-Rotach-type asymptotics around the largest zero (the soft edge asymptotics) for some classes of polynomials satisfying three-term recurrence relations with exponentially increasing coefficients. As special cases, our results include this type of asymptotics for q^{-1}-Hermite polynomials of Askey, Ismail and Masson, q-Laguerre polynomials, and the Stieltjes-Wigert polynomials. We also introduce a one parameter family of solutions to the q-difference equation of the Ramanujan function.