On the asymptotic expansions of products related to the Wallis, Weierstrass and Wilf formulas

Abstract

For all integers $n\geq1$, let \begin{align*} W_n(p,q)=\prod_{j=1}^{n}\left\{e^{-p/j}\left(1+\frac{p}{j}+\frac{q}{j^2}\right)\right\} \end{align*} and \begin{align*} R_n(p, q)=\prod_{j=1}^{n}\left\{e^{-p/(2j-1)}\left(1+\frac{p}{2j-1}+\frac{q}{(2j-1)^2}\right)\right\}, \end{align*} where $p$, $q$ are complex parameters. The infinite product $W_{\infty}(p,q)$ includes the Wallis and Wilf formulas, and also the infinite product definition of Weierstrass for the gamma function, as special cases. In this paper, we present asymptotic expansions of $W_n(p,q)$ and $R_n(p, q)$ as $n\to\infty$. In addition, we also establish asymptotic expansions for the Wallis sequence.