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Completely monotone generalized Mittag-Leffler functions. (English) Zbl 0843.60024
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.

60E99Distribution theory in probability theory
33C99Hypergeometric functions
60A99Foundations of probability theory