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

TY - JOUR

T1 - A one-parameter family of pick functions defined by the gamma function and related to the volume of the unit ball in n-space

AU - Berg, Christian

AU - Pedersen, Henrik Laurberg

PY - 2011/6

Y1 - 2011/6

N2 - We show that Fa(x) =ln Γ(x + 1)/x ln(ax) 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 f(x) =(πx/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,∞[. In particular, f(n) = Ω1/(n ln n)n , n ≥ 2, is a Hausdorff moment sequence. Here Ωn is the volume of the unit ball in Euclidean n-space.

AB - We show that Fa(x) =ln Γ(x + 1)/x ln(ax) 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 f(x) =(πx/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,∞[. In particular, f(n) = Ω1/(n ln n)n , n ≥ 2, is a Hausdorff moment sequence. Here Ωn is the volume of the unit ball in Euclidean n-space.

KW - Faculty of Science

KW - Former LIFE faculty

U2 - 10.1090/S0002-9939-2010-10636-6

DO - 10.1090/S0002-9939-2010-10636-6

M3 - Journal article

SN - 0002-9939

VL - 139

SP - 2121

EP - 2132

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 6

ER -