Remark on a Mittag-Leffler function of Le Roy type

TL;DR

This paper investigates the conditions under which a Mittag-Leffler function of Le Roy type is completely monotonic on the negative half-line, proposing a conjecture about its associated random variable's divisibility properties.

Contribution

It provides necessary and sufficient conditions for complete monotonicity and introduces a conjecture relating to the infinite divisibility of the underlying random variable.

Findings

Identifies conditions for complete monotonicity of the function.

Proposes a conjecture on the logarithmic infinite divisibility.

Advances understanding of the probabilistic properties of Mittag-Leffler functions.

Abstract

We give some necessary and some sufficient conditions for the complete monotonicity on the negative half-line of a Mittag-Leffler function of Le Roy type. It is conjectured that the underlying positive random variable, when it exists, must be logarithmically infinitely divisible.