Jacobi polynomial moments and products of random matrices (Q2827374)

The authors study the eigenvalue distribution for products of random matrices of the form \(Y^* Y\) with \(Y = G_r \ldots G_{s+1} T_s\ldots T_1\), where \(T_1,\ldots,T_s\) are truncated Haar matrices, \(G_{s+1},\ldots,G_r\) are Ginibre matrices, all matrices being independent. The authors introduce certain sequences defined in terms of Jacobi polynomials and show that there is a family of compactly supported distributions solving the Hausdorff moment problem for these sequences. The resulting distributions are free multiplicative convolutions of the so-called Fuss-Catalan and Raney distributions (whose moments are the Fuss-Catalan and Raney numbers known in combinatorics).