Fuss-Catalan numbers in noncommutative probability

We prove that if p, r ∈ R, p ≥ 1 and 0 ≤ r ≤ p then the Fuss-Catalan sequence ( mp+r m ) r mp+r 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 that if 0 ≤ 2r ≤ p and r + 1 ≤ p then μ(p, r) is ⊞-infinitely divisible. As a by-product, we show that the sequence m m m! is positive definite and the corresponding probability measure is ⊠-infinitely divisible. 2010 Mathematics Subject Classification: Primary 46L54; Secondary 44A60, 60C05