Plancherel–Rotach asymptotics for some q-orthogonal polynomials with complex scalings

Abstract

In this work we study the Plancherel–Rotach type asymptotics for Stieltjes–Wigert, q-Laguerre and Ismail–Masson orthogonal polynomials with complex scalings. The main terms of the asymptotics for Stieltjes–Wigert and q-Laguerre polynomials (Ismail–Masson polynomials) contain Ramanujan function A q ( z ) for scaling parameters above the vertical line R ( s ) = 2 ( R ( w ) = 1 2 ); the main terms of the asymptotics involve theta function for scaling parameters in the vertical strip 0 < R ( s ) < 2 ( 0 < R ( w ) < 1 2 ). When scaling parameters in the vertical strips, the number theoretical properties of scaling parameters completely determine the orders of the error terms. These asymptotic formulas may provide some insights to new random matrix models and also add a new link between special functions and number theory.