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

TL;DR

This paper investigates special functions related to the volumes of unit balls in n-dimensional space, establishing their properties as Pick and Stieltjes functions and their implications for monotonicity and moment sequences.

Contribution

It introduces new functions connected to geometric volumes, proves their classification as Pick and Stieltjes functions, and demonstrates their monotonicity and moment sequence properties.

Findings

F_a(x) is a Pick function for a ≥ 1 with an explicit integral representation.

ln f(x+1) is a Stieltjes function and f(x+1) is completely monotonic on (0, ∞).

The sequence Ω_n^{1/(n ln n)} forms a Hausdorff moment sequence.

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