Asymptotic formulas for the gamma function constructed by bivariate means

2014/09/23 by Zhen-Hang Yang, Yang, Zhen-Hang
Mathematics · #11B83 #26E60 #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Primary 33B15 #Secondary 26D15

paper · pdf · doi:10.48550/arxiv.1409.6413

openalex publication_date 2014/09/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let K,M,N denote three bivariate means. In the paper, the author prove the asymptotic formulas for the gamma function have the form of% Γ( x+1) \thicksim √(2π)M( x+θ,x+1-θ) ^{K( x+ε,x+1-ε)} e^{-N( x+σ,x+1-σ)} % or% Γ( x+1) \thicksim √(2π)M( x+θ,x+σ) ^{K( x+ε,x+1-ε)} e^{-M( x+θ,x+σ)} % as x→ ∞ , where ε,θ,σ are fixed real numbers. This idea can be extended to the psi and polygamma functions. As examples, some new asymptotic formulas for the gamma function are presented.

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