Plancherel-Rotach Asymptotics for $q$-Series

Abstract

In this work we study the Plancherel-Rotach type asymptotics for selected $q$-series and $q$-orthogonal polynomials with complex scalings. The $q$-series we cover are Euler's $q$-exponential, Ramanujan function, Jackson's $q$-Bessel function of second kind, Ismail-Masson orthogonal polynomials, Stieltjes-Wigert polynomials and $q$-Laguerre polynomials. For a fixed $q$ with $0<q<1$, in each case the main term of the asymptotic formulas may contain Ramanujan function or theta function depending on the value of scaling parameter. Furthemore, when the scaling parameter is in certain strip of the complex plane, its number theoretical property completely determines the order of the error term. In each cases, we also investigate the asymptotic behavior of the mentioned $q$-series when $q$ approaching 1 in a restricted manner. These asymptotic formulas may provide insights to new random matrix models.