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Pollard used contour integration to show that the Mittag-Leffler function isthe Laplace transform of a positive function, thereby proving that it iscompletely monotone. He also cited personal communication by Feller of adiscovery of the result by "methods of probability theory". In his publishedwork, Feller used the two-dimensional Laplace transform of a bivariatedistribution to derive the Pollard result. But both approaches may be describedas analytic, despite the occurrence of the stable distribution in Feller'sstarting point and in the Pollard result itself. We adopt a Bayesianprobabilistic approach that assigns a prior distribution to the scale parameterof the stable distribution. We present Feller's method as a particular instanceof such assignment. The Bayesian framework enables generalisation of thePollard result. This leads to a novel integral representation of theMittag-Leffler function as well as a variant arising from polynomial tilting ofthe stable density.

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