Résumé : In this expository talk we present a constructive approach to generate exact solutions of the one-dimensional Stieltjes moment problem, with moments expressible via factorials and/or gamma functions. We will mainly concentrate on continuous solutions. The main ingredient of this method is a systematic use of the inverse Mellin transform and of a generalization of hypergeometric function called the Meijer G function, fully implemented in computer algebra systems. The essential property inherent in these tools is the Mellin (i.e. multiplicative) convolution. It allows one to produce exact solutions for a large family of various moment sets, especially those being combinatorial sequences. We enumerate various, older and more recent, criteria for positivity as well as for the uniqueness of so obtained solutions. Our method permits one to bypass (in many cases) an involved question of verifications of these criteria. Amongst concrete examples reviewed here we mention the explicit construction of the so called Stieltjes classes of non-unique solutions via polynomial killers, the problem of non-unique Lévy stable distributions and the construction of moment filters. Finally, a possible application of this methodology to purely discrete distributions will be briefly exposed.

Dernière modification : Monday 18 March 2024

Contact pour cette page : Cyril.Banderier at lipn.univ-paris13.fr