Fuss-Catalan numbers in noncommutative probability

<jats:p> We prove that if <jats:inline-formula> <jats:tex-math>p,r\in{R}, p\ge1</jats:tex-math> </jats:inline-formula> and <jats:inline-formula> <jats:tex-math>0le rle p</jats:tex-math> </jats:inline-formula> then the Fuss-Catalan sequence <jats:inline-formula> <jats:tex-math>\binom{mp+r}m\frac{r}{mp+r}</jats:tex-math> </jats:inline-formula> is positive definite. We study the family of the corresponding probability measures <jats:inline-formula> <jats:tex-math>\mu(p,r)</jats:tex-math> </jats:inline-formula> on <jats:inline-formula> <jats:tex-math>{R}</jats:tex-math> </jats:inline-formula> from the point of view of noncommutative probability. For example, we prove that if <jats:inline-formula> <jats:tex-math>0le 2rle p</jats:tex-math> </jats:inline-formula> and <jats:inline-formula> <jats:tex-math>r+1le p</jats:tex-math> </jats:inline-formula> then <jats:inline-formula> <jats:tex-math>\mu(p,r)</jats:tex-math> </jats:inline-formula> is <jats:inline-formula> <jats:tex-math>\boxplus</jats:tex-math> </jats:inline-formula> -infinitely divisible. As a by-product, we show that the sequence <jats:inline-formula> <jats:tex-math>\frac{m^m}{m!}</jats:tex-math> </jats:inline-formula> is positive definite and the corresponding probability measure is <jats:inline-formula> <jats:tex-math>\boxtimes</jats:tex-math> </jats:inline-formula> -infinitely divisible. </jats:p>