Completely Monotonic Gamma Ratio and Infinitely Divisible H-Function of Fox

By Dmitrii Karp, E. Prilepkina. 2015

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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.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/4f87c74d32a732a3b2293f13c49a67ad0d656eec