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

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• 作者：D. B. Karp ; E. G. Prilepkina
• 关键词：Gamma function ; Completely monotonic functions ; Meijer’s G ; function ; Fox’s H ; function ; Infinite divisibility
• 刊名：Computational Methods and Function Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：16
• 期：1
• 页码：135-153
• 全文大小：502 KB
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• 作者单位：D. B. Karp (1) (3)
  E. G. Prilepkina (1) (2)
  
  1. Far Eastern Federal University, 8 Sukhanova street, Vladivostok, 690950, Russia
  3. Universidad del Atlántico, Km 7 Antigua via, Puerto Colombia, Colombia
  2. Institute of Applied Mathematics, FEBRAS, 7 Radio Street, Vladivostok, 690041, Russia
  
• 刊物主题：Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable;
• 出版者：Springer Berlin Heidelberg
• ISSN：2195-3724

文摘

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.