Asymptotic formulas for the gamma function constructed by bivariate means

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*} Γ\left( x+1\right) \thicksim \sqrt{2π}M\left( x+θ,x+1-θ\right) ^{K\left( x+ε,x+1-ε\right) }e^{-N\left( x+σ,x+1-σ\right) } \end{equation*}% or% \begin{equation*} Γ\left( x+1\right) \thicksim \sqrt{2π}M\left( x+θ,x+σ\right) ^{K\left( x+ε,x+1-ε\right) }e^{-M\left( x+θ,x+σ\right) } \end{equation*}% as $x\rightarrow \infty$, 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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