A Pick function related to the sequence of volumes of the unit ball in n-space

Abstract

We show that F_a(x)=\frac{\ln \Gamma (x+1)}{x\ln(ax)} is a Pick function for a\ge 1 and find its integral representation. We also consider the function f(x)=(\frac{\pi^{x/2}}{\Gamma(1+x/2)})^{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