A new property of the Wallis power function.

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. [ABSTRACT FROM AUTHOR]