Plancherel-Rotach Asymptotics for $q$-Laguerre Orthogonal Polynomials with Complex Scaling

In this work we study the Plancherel-Rotach type asymptotics for $q$-Laguerre 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 function 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 insights to some new random matrix model and add a new link between special functions and number theory.