A ONE-PARAMETER FAMILY OF PICK FUNCTIONS DEFINED BY THE GAMMA FUNCTION AND RELATED TO THE VOLUME OF THE UNIT BALL IN n-SPACE

We show that F(a)(x) = ln Gamma(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) = (pi(x/2)/Gamma(1 + x/2))(1/(x ln x)) and show that In f(x + 1) is a Stieltjes function and that f(x + 1) is completely monotonic on ]0, infinity[. In particular, f(n) = Omega(1/(n ln n))(n), n >= 2, is a Hausdorff moment sequence. Here Omega(n) is the volume of the unit ball in Euclidean n-space.