Moment problems related to Bernstein functions

Simon, Thomas

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 3, pp. 577-594.

Nous donnons une preuve simple du caractère indéterminé sur la demi-droite de la suite de moments entiers ${(n!)}^t$ pour $t>2,$ à l’aide de la condition de Lin. Sous une hypothèse d’auto-décomposabilité logarithmique, la méthode s’étend à des suites de puissances de moments entiers définis comme la factorielle croissante d’une fonction de Bernstein donnée, et plus généralement à d’autres suites infiniment divisibles de moments entiers. Nous donnons aussi une preuve très courte du caractère infiniment divisible de toutes les suites de moments entiers récemment étudiées dans [16] et en particulier de la suite de Fuss–Catalan.

We give a simple proof of the moment-indeterminacy on the half-line of the sequence ${(n!)}^t$ for $t>2,$ using Lin’s condition. Under a logarithmic self-decomposability assumption, the method conveys to power moment sequences defined as the rising factorials of a given Bernstein function, and to more general infinitely divisible moment sequences. We also provide a very short proof of the infinite divisibility of all the integer moment sequences recently investigated in [16], including Fuss–Catalan’s.

Classification : 44A60, 60E05, 60G51
Mots clés : Bernstein function, Fuss–Catalan number, Moment problem, Moment sequence, Remainder

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