Completely Monotonic Gamma Ratio and Infinitely Divisible H-Function of Fox
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Completely Monotonic Gamma Ratio and Infinitely Divisible H-Function of Fox D. B. Karp1,3 · E. G. Prilepkina1,2
Received: 28 January 2015 / Revised: 17 April 2015 / Accepted: 10 May 2015 © Springer-Verlag Berlin Heidelberg 2015
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. Keywords Gamma function · Completely monotonic functions · Meijer’s G-function · Fox’s H -function · Infinite divisibility Mathematics Subject Classfication
33B15 · 33C60
Communicated by Stephan Ruscheweyh.
B B
D. B. Karp [email protected] E. G. Prilepkina [email protected]
1
Far Eastern Federal University, 8 Sukhanova street, Vladivostok 690950, Russia
2
Institute of Applied Mathematics, FEBRAS, 7 Radio Street, Vladivostok 690041, Russia
3
Universidad del Atlántico, Km 7 Antigua via, Puerto Colombia, Colombia
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D. B. Karp, E. G. Prilepkina
1 Introduction Recall that a non-negative function f defined on (0, ∞) is called completely monotonic (c.m.) if it has derivatives of all orders and (−1)n f (n) (x) ≥ 0 for n ≥ 1 and x > 0 [18, Def. 1.3]. This inequality is known to be strict unless f is a constant. By the celebrated Bernstein theorem, a function is completely monotonic if and only if it is the Laplace transform of a non-negative measure [18, Thm. 1.4]. The above definition implies the following equivalences f is c.m. on (0, ∞) ⇔ f ≥ 0 and − f is c.m. on (0, ∞) ⇔ − f is c.m. on (0, ∞) and lim f (x) ≥ 0. x→∞
(1)
A positive function f is said to be logarithmically completely monotonic (l.c.m.) if −(log f ) is completely monotonic [18, Def. 5.8]. According to (1) f is l.c.m. on (0, ∞) ⇔ (− log f (x)) ≥ 0 and (log f ) is c.m. on (0, ∞) lim (− log f (x)) ≥ 0. ⇔ (log f ) is c.m. on (0, ∞) and x→∞
(2) The class of l.c.m. functions is a proper subset of the class of c.m. functions. Their importance stems from the fact that they represent Laplace transforms of infinitely divisible probability distributions, see [18, Thm. 5.9] and [17, Sec. 51]. The study of complete monotonicity of the ratio U (x) =
p (x + ai ) , (x + bi )
(3)
i=1
where stands for Euler’s gamma function and p ≥ 1 has been initiated by Bustoz and Ismail who demonstrated in their 1986 paper [4] that for p
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