Fuss-Catalan numbers in noncommutative probability

  • Wojciech Mlotkowski

    Mathematical Institute University of Wroclaw Pl. Grunwaldzki 2/4 50-384 Wroclaw Poland
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We prove that if p,rR,p1p,r\in{R}, p\ge1 and 0lerlep0le rle p then the Fuss-Catalan sequence (mp+rm)rmp+r\binom{mp+r}m\frac{r}{mp+r} is positive definite. We study the family of the corresponding probability measures μ(p,r)\mu(p,r) on R{R} from the point of view of noncommutative probability. For example, we prove that if 0le2rlep0le 2rle p and r+1lepr+1le p then μ(p,r)\mu(p,r) is \boxplus-infinitely divisible. As a by-product, we show that the sequence mmm!\frac{m^m}{m!} is positive definite and the corresponding probability measure is \boxtimes-infinitely divisible.

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Wojciech Mlotkowski, Fuss-Catalan numbers in noncommutative probability. Doc. Math. 15 (2010), pp. 939–955

DOI 10.4171/DM/318