  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.
