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 can be considered as a Pick function when a > 1, i.e. extends to a holomorphic function mapping the upper half-plane into itself. We also consider the function and show that In f(x+1) is a Stieltjes function and that f(x + 1) is completely monotonic on ]0, ∞[. In particular, f(n) = Ω 1/(n ln n) n , n ≥ 2, is a Hausdorff moment sequence. Here Ω n is the volume of the unit ball in Euclidean n-space.