Cauchy Biorthogonal Polynomials

TL;DR

This paper studies biorthogonal polynomials related to a specific approximation scheme, focusing on their properties, zeroes, recurrence relations, and connections to spectral problems and random matrix models.

Contribution

It introduces a new class of biorthogonal polynomials associated with the Cauchy kernel, analyzing their properties and connections to spectral and random matrix theories.

Findings

Zeroes are simple and positive

Polynomials satisfy a four-term recurrence relation

Zeroes are interlaced and satisfy Christoffel-Darboux formulas

Abstract

The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite-Pade' approximation scheme. Associated to any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeroes are simple and positive. We then specialize the kernel to the Cauchy kernel 1/{x+y} and show that the ensuing biorthogonal polynomials solve a four-term recurrence relation, have relevant Christoffel-Darboux generalized formulae, and their zeroes are interlaced. In addition, these polynomial solve a combination of Hermite-Pade' approximation problems to a Nikishin system of order 2. The motivation arises from two distant areas; on one side, in the study of the inverse spectral problem for the peakon solution of the Degasperis-Procesi equation; on the other side, from a random matrix model involving two positive definite random Hermitian matrices. Finally, we show how to characterize these polynomials in term of a Riemann-Hilbert problem.