Fuss-Catalan numbers in noncommutative probability

By Wojciech Mlotkowski. Documenta Mathematica, 2010

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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

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

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Asymptotic distribution of singular values of powers of random matrices (2010, background): “emains an open problem. In Free Probability Theory M(m) p are known as Fuss–Catalan numbers. Combinatorial properties of this sequence have been studied by Nica and Speicher 2006 [8]. Mlotkowski 2009 [7] investigated a family of distributions, say G(m,r), with real m ≥ 0 and 0 ≤ r ≤ m, such that G(m”