Mixed type multiple orthogonal polynomials associated with the modified Bessel functions and products of two coupled random matrices

TL;DR

This paper introduces and analyzes a new class of mixed type multiple orthogonal polynomials linked to modified Bessel functions, with applications to coupled random matrix products and their asymptotic behavior.

Contribution

The work establishes the existence, explicit formulas, and properties of these polynomials, connecting them to biorthogonal functions in random matrix theory and providing a Riemann-Hilbert framework.

Findings

Explicit formulas and integral representations derived.

Connection to biorthogonal functions in random matrix products.

Riemann-Hilbert characterization enabling asymptotic analysis.

Abstract

We consider mixed type multiple orthogonal polynomials associated with a system of weight functions consisting of two vectors. One vector is defined in terms of scaled modified Bessel function of the first kind $I_\mu$ and $I_{\mu+1}$, the other vector is defined in terms of scaled modified Bessel function of the second kind $K_\nu$ and $K_{\nu+1}$. We show that the corresponding mixed type multiple orthogonal polynomials exist. For the special case that each multi-index is on or close to the diagonal, basic properties of the polynomials and their linear forms are investigated, which include explicit formulas, integral representations, differential properties, limiting forms and recurrence relations. It comes out that, for specified parameters, the linear forms of these mixed type multiple orthogonal polynomials can be interpreted as biorthogonal functions encountering in recent studies of products of two coupled random matrices. This particularly implies a Riemann-Hilbert characterization of the correlation kernel, which provides an alternative way for further asymptotic analysis.