Scaled Asymptotics For Some $q$-Series As $q$ Approaching Unit

TL;DR

This paper derives asymptotic formulas for various $q$-series as $q$ approaches 1, extending classical results to a broader class of special functions and orthogonal polynomials.

Contribution

It provides new Plancherel-Rotach type asymptotics for several $q$-series and orthogonal polynomials as $q$ tends to 1, generalizing known results.

Findings

Asymptotic formulas for $A_q(z)$, $J_{ u}^{(2)}(z;q)$, and other $q$-series as $q o 1$

Extension of classical asymptotics to $q$-analogues of special functions

Unified approach to asymptotics for multiple $q$-orthogonal polynomials

Abstract

In this work we investigate Plancherel-Rotach type asymptotics for some $q$-series as $q\to1$. These $q$-series generalize Ramanujan function $A_{q}(z)$; Jackson's $q$-Bessel function $J_{\nu}^{(2)}$(z;q), Ismail-Masson orthogonal polynomials($q^{-1}$-Hermite polynomials) $h_{n}(x|q)$, Stieltjes-Wigert orthogonal polynomials $S_{n}(x;q)$, $q$-Laguerre orthogonal polynomials $L_{n}^{(\alpha)}(x;q)$ and confluent basic hypergeometric series.