Fuss-Catalan Numbers in Noncommutative Probability

Abstract

We prove that if p, r ∈ R, p ≥ 1 and 0 ≤ r ≤ p then the() mp+r Fuss-Catalan sequence (Formula presented) is positive definite. We study the family of the corresponding probability measures µ(p, r) on R from the point of view of noncommutative probability. For example, we prove (Formula presented -infinitely divisible. As a by-product, we show that the sequence (Formula presented) is positive definite and the corresponding probability measure is ⊠-infinitely divisible.