
Romuald Lenczewski

Rafal Salapata
Keywords:
FussNarayana numbers, Narayana polynomials, random matrix, MarchenkoPastur law, free probability
Abstract
It has been shown recently that the limit moments of $W(n)=B(n)B^{*}(n)$, where $B(n)$ is a product of $p$ independent rectangular random matrices, are certain homogeneous polynomials $P_{k}(d_0,d_1, \ldots , d_{p})$ in the asymptotic dimensions of these matrices. Using the combinatorics of noncrossing partitions, we explicitly determine these polynomials and show that they are closely related to polynomials which can be viewed as {\it multivariate FussNarayana polynomials}. Using this result, we compute the moments of $\varrho_{t_1}\boxtimes \varrho_{t_2}\boxtimes\ldots \boxtimes \varrho_{t_m}$ for any positive $t_1,t_2, \ldots , t_m$, where $\boxtimes$ is the free multiplicative convolution in free probability and $\varrho_{t}$ is the MarchenkoPastur distribution with shape parameter $t$.