A new asymptotic expansion of a ratio of two gamma functions and complete monotonicity for its remainder

Abstract:

In this paper, we establish a new asymptotic expansion of a ratio of two gamma functions, that is, as $x\rightarrow \infty$, \begin{equation*} \left [ \frac {\Gamma \left ( x+u\right ) }{\Gamma \left ( x+v\right ) }\right ] ^{1/\left ( u-v\right ) }\thicksim \left ( x+\sigma \right ) \exp \left [ \sum _{k=1}^{m}\frac {B_{2n+1}\left ( \rho \right ) }{wn\left ( 2n+1\right ) }\left ( x\!+\!\sigma \right ) ^{-2k}\!+\!R_{m}\left ( x;u,v\right ) \right ] , \end{equation*} where $u,v\in \mathbb {R}$ with $w=u-v\neq 0$ and $\rho =\left ( 1-w\right ) /2$, $\sigma =\left ( u+v-1\right ) /2$, $B_{2n+1}\left ( \rho \right )$ are the Bernoulli polynomials. We also prove that the function $x\mapsto \left ( -1\right ) ^{m}R_{m}\left ( x;u,v\right )$ for $m\in \mathbb {N}$ is completely monotonic on $\left ( -\sigma ,\infty \right )$ if $\left \vert u-v\right \vert <1$, which yields an explicit bound for $\left \vert R_{m}\left ( x;u,v\right ) \right \vert$ and some new inequalities.