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We investigate conditions for logarithmic complete monotonicity of a quotientof two products of gamma functions, where the argument of each gamma functionhas different scaling factor. We give necessary and sufficient conditions interms of nonnegativity of some elementary function and more practicalsufficient conditions in terms of parameters. Further, we study therepresenting measure in Bernstein's theorem for both equal and non-equalscaling factors. This leads to conditions on parameters under which Meijer's$G$-function or Fox's $H$-function represents an infinitely divisibleprobability distribution on positive half-line. Moreover, we present newintegral equations for both $G$-function and $H$-function. The results of thepaper generalize those due to Ismail (with Bustoz, Muldoon and Grinshpan) andAlzer who considered previously the case of unit scaling factors.

Completely monotonic gamma ratio and infinitely divisible H-function of Fox