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

TL;DR

This paper derives asymptotic expansions for products related to the Wallis, Weierstrass, and Wilf formulas, generalizing classical results and providing new insights into their behavior as the index tends to infinity.

Contribution

The paper introduces new asymptotic expansions for products involving parameters p and q, extending classical formulas like Wallis and Weierstrass to more general cases.

Findings

Asymptotic expansions for W_n(p,q) as n→∞

Asymptotic expansions for R_n(p,q) as n→∞

Asymptotic behavior of the Wallis sequence

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.