## 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 $$\mathbb{R}_+$$ (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:

 60E99 Distribution theory 33C99 Hypergeometric functions 60A99 Foundations of probability theory