Asymptotic formulas for the gamma function constructed by bivariate means

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.