Plancherel-Rotach asymptotics for some \(q\)-orthogonal polynomials with complex scalings (Q927675)

After recalling some properties of \(q\)-series and generalized irrational measures the author presents the Plancherel-Rotach asymptotic formulas for Stieltjes-Wigert, \(q\)-Laguerre and Ismail-Masson orthogonal polynomials under complex scaling depending on two real parameters. They are orthogonal polynomials associated with indeterminant moment problem. According to the different cases where the two parameters of scaling are positive, rational or irrational the author shows that the main term of asymptotics may contain Ramanujan function \(A_q(z)\) or teta function. The obtained results gain some insight to the related random matrix model and number theory.