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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Christian Berg and Henrik L. Pedersen
Journal: Proc. Amer. Math. Soc. 139 (2011), 2121-2132
MSC (2010): Primary 33B15; Secondary 30E20, 30E15
Published electronically: November 19, 2010
MathSciNet review: 2775390
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Abstract: We show that

$\displaystyle F_a(x)=\frac{\ln \Gamma (x+1)}{x\ln(ax)} $

can be considered as a Pick function when $ a\ge 1$, i.e. extends to a holomorphic function mapping the upper half-plane into itself. We also consider the function

$\displaystyle f(x)=\left(\frac{\pi^{x/2}}{\Gamma(1+x/2)}\right)^{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.


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Additional Information

Christian Berg
Affiliation: Institute of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 København Ø, Denmark
Email: berg@math.ku.dk

Henrik L. Pedersen
Affiliation: Department of Basic Sciences and Environment, Faculty of Life Sciences, University of Copenhagen, Thorvaldsensvej 40, DK-1871 Frederiksberg C, Denmark
Email: henrikp@dina.kvl.dk

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10636-6
PII: S 0002-9939(2010)10636-6
Keywords: Gamma function, completely monotonic function, Pick function
Received by editor(s): December 10, 2009
Received by editor(s) in revised form: June 9, 2010
Published electronically: November 19, 2010
Additional Notes: Both authors acknowledge support by grant 272-07-0321 from the Danish Research Council for Nature and Universe.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2010 American Mathematical Society