zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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.

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