Strong asymptotic of Cauchy biorthogonal polynomials and orthogonal polynomials with varying measure

TL;DR

This paper derives the strong asymptotic behavior of Cauchy biorthogonal polynomials supported on non-intersecting intervals, linking them to orthogonal polynomials with varying measures under Szegő's condition.

Contribution

It provides the first detailed asymptotic analysis of Cauchy biorthogonal polynomials in this setting, connecting them to orthogonal polynomials with varying measures.

Findings

Established strong asymptotics for Cauchy biorthogonal polynomials

Linked biorthogonal polynomials to orthogonal polynomials with varying measures

Utilized Szegő's condition to derive asymptotic results

Abstract

We give the strong asymptotic of Cauchy biorthogonal polynomials under the assumption that the defining measures are supported on non intersecting intervals of the real line and satisfy Szeg\H{o}'s condition. The biorthogonal polynomials are connected with certain mixed type Hermite-Pad\'e polynomials, which verify full orthogonality relations with respect to some varying measures. Thus, the strong asymptotic of orthogonal polynomials with respect to varying measures plays a key role in the study.