Discrete analogues of Laplace's approximation

Abstract

An asymptotic formula is derived for the sum

I_n(1|q):=Σⁿ_k=0f_n(k)q^g_n(k)

as n→∞, where f_n(k) and g_n(k) are functions defined on nonnegative integers and 0<q<1. This formula is a discrete analogue of Laplace's approximation for integrals. Corresponding results are also provided for the more general sum

I_n(z|q):=Σⁿ_k=0f_n(k)q^g_n(k)z^k

which is typically an nth order polynomial. The results obtained are then used to give asymptotic formulas for the q⁻¹-Hermite polynomial h_n(x|q), the Stieltjes–Wigert polynomial S_n(x; q) and the q-Laguerre polynomial L^α_n(x; q).