Plancherel-Rotach Asymptotics for Stieltjes-Wigert Orthogonal Polynomials with Complex Scaling

In this work we study the Plancherel-Rotach type asymptotics for Stieltjes-Wigert orthogonal polynomials with complex scaling. The main term of the asymptotics contains Ramanujan function $A_{q}(z)$ for the scaling parameter on the vertical line $\Re(s)=2$, while the main term of the asymptotics involves the theta functions for the scaling parameter in the strip $0<\Re(s)<2$. In the latter case the number theoretical property of the scaling parameter completely determines the order of the error term. $ $ These asymptotic formulas may provide some insights to some new random matrix model and also add a new link between special functions and number theory.