On Feller, Pollard and the Complete Monotonicity of the Mittag-Leffler Function $E_α(-x)$

Abstract

Pollard used contour integration to show that the Mittag-Leffler function is the Laplace transform of a positive function, thereby proving that it is completely monotone. He also cited personal communication by Feller of a discovery of the result by "methods of probability theory". In his published work, Feller used the two-dimensional Laplace transform of a bivariate distribution to derive the Pollard result. But both approaches may be described as analytic, despite the occurrence of the stable distribution in Feller's starting point and in the Pollard result itself. We adopt a Bayesian probabilistic approach that assigns a prior distribution to the scale parameter of the stable distribution. We present Feller's method as a particular instance of such assignment. The Bayesian framework enables generalisation of the Pollard result. This leads to a novel integral representation of the Mittag-Leffler function as well as a variant arising from polynomial tilting of the stable density.