On The Chaotic Asymptotics of Ramanujan's Entire Function $A_q(z)$

Abstract

We will use a discrete analogue of the classical \emph{Laplace} method to show that for infinitely many positive integers $n$, the main term of the asymptotic expansions of scaled confluent basic hypergeometric functions, including the Ramanujan's entire function $A_{q}(z)$, could be expressed in $θ$-functions, and the error term depends on the ergodic property of certain real scaling parameter.