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

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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 Aq(z) for the scaling parameter on the vertical line (s)=2, while the main term of the asymptotics involves the theta functions for the scaling parameter in the strip 0<(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.