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

2015/11/30 by C. P. Chen, C. -P. Chen, Chen, C. -P. +2
Mathematics · #33B15 #41A60 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #math.CA #msc:33B15 #msc:41A60

paper · pdf · doi:10.48550/arxiv.1511.09217

15 pages, 0 figures

arxiv created 2015/11/30 · openalex publication_date 2015/11/30 · arxiv updated 2015/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For all integers n≥1, let W_n(p,q)=∏_j=1ⁿ\e^-p/j(1+(p)/(j)+(q)/(j²))\ and R_n(p, q)=∏_j=1ⁿ\e^-p/(2j-1)(1+(p)/(2j-1)+(q)/((2j-1)²))\, where p, q are complex parameters. The infinite product W_∞(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→∞. In addition, we also establish asymptotic expansions for the Wallis sequence.

Related

• Generalising the Wallis Product
• Generalizing Wallis formula
• An asymptotic expansion for a ratio of products of gamma functions
• Asymptotic expansions for ratios of products of gamma functions
• On asymptotics, Stirling numbers, Gamma function and polylogs