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arXiv · 0912.2185

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

Christian Berg·Henrik L. Pedersen
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Abstract

We show that F_a(x)=\frac{\ln Γ(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{π^{x/2}}{Γ(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)=Ω_n^{1/(n\ln n)},n\ge 2 is a Hausdorff moment sequence. Here Ω_n is the volume of the unit ball in Euclidean n-space

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Christian Berg, Henrik L. Pedersen. 2009-12-11. A Pick function related to the sequence of volumes of the unit ball in n-space. https://arxiv.org/abs/0912.2185

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