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*} 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 $x\to \infty$, 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.

21 pages

References in corpus (1)

• Integral representations and complete monotonicity related to the remainder of Burnside's formula for the gamma function