### Multivariate Fuss-Narayana Polynomials and their Application to Random Matrices

#### 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 Fuss-Narayana 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 Marchenko-Pastur distribution with shape parameter $t$.

#### Keywords

Fuss-Narayana numbers, Narayana polynomials, random matrix, Marchenko-Pastur law, free probability