Abstract
We investigate conditions for logarithmic complete monotonicity of a quotient of two products of gamma functions, where the argument of each gamma function has a different scaling factor. We give necessary and sufficient conditions in terms of non-negativity of some elementary functions and some more practical sufficient conditions in terms of parameters. Further, we study the representing measure in Bernstein’s theorem for both equal and non-equal scaling factors. This leads to conditions on parameters under which Meijer’s G-function or Fox’s H-function represents an infinitely divisible probability distribution on the positive half-line. Moreover, we present new integral equations for both G-function and H-function. The results of the paper generalize those due to Ismail (with Bustoz, Muldoon and Grinshpan) and Alzer who considered previously the case of unit scaling factors.
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Acknowledgments
We thank the anonymous referee for correcting a mistake in the original version of the paper and numerous useful suggestions that helped to improve the exposition substantially. This work has been supported by the Russian Science Foundation under project 14-11-00022.
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Communicated by Stephan Ruscheweyh.
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Karp, D.B., Prilepkina, E.G. Completely Monotonic Gamma Ratio and Infinitely Divisible H-Function of Fox. Comput. Methods Funct. Theory 16, 135–153 (2016). https://doi.org/10.1007/s40315-015-0128-9
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DOI: https://doi.org/10.1007/s40315-015-0128-9
Keywords
- Gamma function
- Completely monotonic functions
- Meijer’s G-function
- Fox’s H-function
- Infinite divisibility