Asymptotics of a ${}_3{F}_2$ polynomial associated with the Catalan-Larcombe-French sequence

Abstract

The large $n$ behaviour of the hypergeometric polynomial $$\FFF{-n}{\sfrac12}{\sfrac12}{\sfrac12-n}{\sfrac12-n}{-1}$$ is considered by using integral representations of this polynomial. This ${}_3{F}_2$ polynomial is associated with the Catalan-Larcombe-French sequence. Several other representations are mentioned, with references to the literature, and another asymptotic method is described by using a generating function of the sequence. The results are similar to those obtained by Clark (2004) who used a binomial sum for obtaining an asymptotic expansion.

arXiv comment10 pages, 1 figure. Accepted for publication in {\em Analysis and Applications}