Asymptotic formulas for the gamma function constructed by bivariate means

TL;DR

This paper derives new asymptotic formulas for the gamma function using bivariate means, extending the approach to psi and polygamma functions, and provides concrete examples of these formulas.

Contribution

The paper introduces novel asymptotic formulas for the gamma function based on bivariate means, expanding the theoretical framework and including extensions to related functions.

Findings

New asymptotic formulas for gamma function derived

Formulas extend to psi and polygamma functions

Examples illustrating the formulas are provided

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% \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta,x+1-\theta \right) ^{K\left( x+\epsilon ,x+1-\epsilon \right) }e^{-N\left( x+\sigma ,x+1-\sigma \right) } \end{equation*}% or% \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\sigma \right) ^{K\left( x+\epsilon ,x+1-\epsilon \right) }e^{-M\left( x+\theta ,x+\sigma \right) } \end{equation*}% as $x\rightarrow \infty $, where $\epsilon ,\theta ,\sigma $ 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.