Global Asymptotics of Stieltjes-Wigert Polynomials

TL;DR

This paper derives asymptotic formulas for Stieltjes-Wigert polynomials across the entire complex plane using $q$-Airy functions, and explores their relation to classical Airy functions as $q$ approaches 1.

Contribution

It provides comprehensive asymptotic formulas for Stieltjes-Wigert polynomials in the complex plane and links $q$-Airy functions to classical Airy functions in the limit.

Findings

Asymptotic formulas valid in all regions of the complex plane.

Use of $q$-Airy functions as approximants for the polynomials.

Limiting relation between $q$-Airy and Airy functions as $q o 1$.

Abstract

Asymptotic formulas are derived for the Stieltjes-Wigert polynomials $S_n(z;q)$ in the complex plane as the degree $n$ grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc centered at the origin; the two regions together cover the whole plane. In each region, the $q$-Airy function $A_q(z)$ is used as the approximant. For real $x> 1/4$, a limiting relation is also established between the $q$-Airy function $A_q(x)$ and the ordinary Airy function $\mathrm{Ai}(x)$ as $q \to 1$.