  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.
