Plancherel–Rotach asymptotics for certain basic hypergeometric series

Abstract

In this work we study the chaotic and periodic asymptotics for the confluent basic hypergeometric series. For a fixed q ∈ ( 0 , 1 ) , the asymptotics for Euler's q-exponential, q-Gamma function Γ q ( x ) , q-Airy function of K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Ramanujan function ( q-Airy function), Jackson's q-Bessel function of second kind, Ismail–Masson orthogonal polynomials ( q −1 -Hermite polynomials), Stieltjes–Wigert polynomials, q-Laguerre polynomials could be derived as special cases.