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

 

Completely monotonic functions involving the gamma and $q$-gamma functions


Authors: Arcadii Z. Grinshpan and Mourad E. H. Ismail
Journal: Proc. Amer. Math. Soc. 134 (2006), 1153-1160
MSC (2000): Primary 33B15; Secondary 26A48
Published electronically: September 28, 2005
MathSciNet review: 2196051
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Abstract: We give an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic. Each such function gives rise to an infinitely divisible probability distribution. Other similar results are also obtained for specific combinations of the gamma and $q$-gamma functions.


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Additional Information

Arcadii Z. Grinshpan
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
Email: azg@math.usf.edu

Mourad E. H. Ismail
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: ismail@math.ucf.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08050-0
PII: S 0002-9939(05)08050-0
Keywords: Gamma function, $q$-gamma function, completely monotonic functions, inequalities.
Received by editor(s): April 12, 2004
Received by editor(s) in revised form: October 28, 2004, and November 9, 2004
Published electronically: September 28, 2005
Additional Notes: The second author’s research was partially supported by NSF grant DMS 99-70865.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2005 American Mathematical Society