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Zbl 0843.60024
Schneider, W.R.
Completely monotone generalized Mittag-Leffler functions. (English)
[J] Expo. Math. 14, No.1, 3-16 (1996). ISSN 0723-0869

Summary: The generalized Mittag-Leffler function $$F_{\alpha, \beta}(t) = \Gamma(\beta) \sum^\infty_{k = 0} {(-t)^k \over \Gamma(\alpha k + \beta)}, \qquad t \geq 0, \quad \alpha > 0, \quad \beta > 0,$$ is shown to be completely monotone iff the parameters $\alpha$ and $\beta$ satisfy $0 < \alpha \leq 1$, $\beta \geq \alpha$. As $F_{\alpha, \beta} (0) = 1$, the if-part is equivalent to the statement that $F_{\alpha, \beta}$ is the Laplace transform of a probability measure $\mu_{\alpha, \beta}$ supported by $\bbfR_+$ (Bernstein's theorem). Apart from the trivial case $\alpha = \beta = 1$ these measures are absolutely continuous with respect to the Lebesgue measure, and explicit representations of the associated densities are obtained.

MSC 2000:: *60E99 Distribution theory in probability theory
33C99 Hypergeometric functions
60A99 Foundations of probability theory

Keywords: Bernstein's theorem; Laplace transform of a probability measure

Cited in: Zbl 0964.33011

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