Random Matrices: Theory and Applications

A survey is given on sequences of orthogonal polynomials related to Stieltjes functions satisfying a Riccati type differential equation with polynomial coefficients — the so-called Laguerre–Hahn class. The main goal is to describe analytical aspects, focusing on differential equations for those orthogonal polynomials, difference and differential equations for the recurrence coefficients, and distributional equations for the corresponding linear functionals.

In this contribution, we consider varying Krall-type polynomials which are orthogonal with respect to a varying discrete Krall-type inner product. Our main goal is to give ladder operators for this family of polynomials as well as to find a second-order differential-difference equation that these polynomials satisfy. We generalize some results that appeared recently in the literature.

We present some elements of the theory of orthogonal polynomials based on matrix decompositions. We focus our attention on discrete linear functionals, and use the Meixner polynomials as a concrete example.

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues, including joint moments, for the case of both ordered and unordered eigenvalues.