Fuss-Catalan numbers in noncommutative probability
Documenta mathematica, Tome 15 (2010), pp. 939-955.

Voir la notice de l'article dans Electronic Library of Mathematics

Summary: We prove that if $p,r\in\{R}, p\ge1$ and $0le rle p$ then the Fuss-Catalan sequence $\binom{mp+r}m\frac{r}{mp+r}$ is positive definite. We study the family of the corresponding probability measures $\mu(p,r)$ on $\{R}$ from the point of view of noncommutative probability. For example, we prove that if $0le 2rle p$ and $r+1le p$ then $\mu(p,r)$ is $\boxplus$-infinitely divisible. As a by-product, we show that the sequence $\frac{m^m}{m!}$ is positive definite and the corresponding probability measure is $\boxtimes$-infinitely divisible.
Classification : 46L54, 44A60, 60C05
Mots-clés : fuss-Catalan numbers, free, Boolean and monotonic convolution 
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     author = {Mlotkowski, Wojciech},
     title = {Fuss-Catalan numbers in noncommutative probability},
     journal = {Documenta mathematica},
     pages = {939--955},
     publisher = {mathdoc},
     volume = {15},
     year = {2010},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/DOCMA_2010__15__a6/}
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Mlotkowski, Wojciech. Fuss-Catalan numbers in noncommutative probability. Documenta mathematica, Tome 15 (2010), pp. 939-955. https://geodesic-test.mathdoc.fr/item/DOCMA_2010__15__a6/