A one-parameter family of Pick functions defined by the Gamma function and related to the volume of the unit ball in $n$-space

Abstract:

We show that \[ F_a(x)=\frac {\ln \Gamma (x+1)}{x\ln (ax)} \] can be considered as a Pick function when $a\ge 1$, i.e. extends to a holomorphic function mapping the upper half-plane into itself. We also consider the function \[ f(x)=\left (\frac {\pi ^{x/2}}{\Gamma (1+x/2)}\right )^{1/(x\ln x)} \] and show that $\ln f(x+1)$ is a Stieltjes function and that $f(x+1)$ is completely monotonic on $]0,\infty [$. In particular, $f(n)=\Omega _n^{1/(n\ln n)},n\ge 2$, is a Hausdorff moment sequence. Here $\Omega _n$ is the volume of the unit ball in Euclidean $n$-space.