Plancherel-Rotach Asymptotics for Ismail-Masson Orthogonal Polynomials with Complex Scaling

In this work we study the Plancherel-Rotach type asymptotics for Ismail-Masson 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 $\Re(s)={1/2}$, while the main term of the asymptotics involves the theta functions for the scaling parameter in the strip $0<\Re(s)<{1/2}$. 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 insights to some new random matrix models and also add a new link between special functions and number theory.