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

Schneider, W. R.

Expo. Math. 14, No. 1, 3-16 (1996).

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

Keywords:

Bernstein’s theorem; Laplace transform of a probability measure