A new property of the Wallis power function

Abstract:

The Wallis power function is defined on $\left ( -\min \left \{ p,q\right \},\infty \right )$ by \begin{equation*} W_{p,q}\left ( x\right ) =\left ( \dfrac {\Gamma \left ( x+p\right ) }{\Gamma \left ( x+q\right ) }\right ) ^{1/\left ( p-q\right ) }\text { if }p\neq q\text { and }W_{p,p}\left ( x\right ) =e^{\psi \left ( x+p\right ) }. \end{equation*} Let $p_{i},q_{i}\in \mathbb {R}$ with $0<\delta _{i}=p_{i}-q_{i}\leq 1$, $\theta _{i}=\left ( 1-\delta _{i}\right ) /2$ for $i=1,2$ and $p_{1}+q_{1}=p_{2}+q_{2}=2\sigma +1$. We prove that, if $q_{1}>q_{2}$ then for any integer $m\in \mathbb {N}$, the function \begin{equation*} x\mapsto \left ( -1\right ) ^{m}\left [ \ln \frac {W_{p_{1},q_{1}}\left ( x\right ) }{W_{p_{2},q_{2}}\left ( x\right ) }-\sum _{k=1}^{m}\frac {a_{2k}\left ( \theta _{1},\theta _{2}\right ) }{\left ( 2k+1\right ) \left ( 2k\right ) \left ( x+\sigma \right ) ^{2k}}\right ] \end{equation*} is completely monotonic on $\left ( -\sigma ,\infty \right )$, where \begin{equation*} a_{2k}\left ( \theta _{1},\theta _{2}\right ) =\frac {B_{2k+1}\left ( \theta _{2}\right ) }{\theta _{2}-1/2}-\frac {B_{2k+1}\left ( \theta _{1}\right ) }{\theta _{1}-1/2}. \end{equation*} This extends and generalizes some known results.