Discrete analogues of Laplace's approximation

Article type: Research Article

Authors: Wang, X.S. | Wong, R.

Affiliations: Joint Advanced Research Center, University of Science and Technology of China-City University of Hong Kong, Renai Road, Dushu Lake Higher Education Town, Suzhou, Jiangshu, 215123, China | Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Abstract: An asymptotic formula is derived for the sum In(1|q):=Σnk=0fn(k)qgn(k) as n→∞, where fn(k) and gn(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 In(z|q):=Σnk=0fn(k)qgn(k)zk which is typically an nth order polynomial. The results obtained are then used to give asymptotic formulas for the q−1-Hermite polynomial hn(x|q), the Stieltjes–Wigert polynomial Sn(x; q) and the q-Laguerre polynomial Lαn(x; q).

Keywords: Laplace's approximation, q-Airy function, q^−1-Hermite polynomial, Stieltjes–Wigert polynomial, q-Laguerre polynomial

Journal: Asymptotic Analysis, vol. 54, no. 3-4, pp. 165-180, 2007