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

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.