Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-type Weight

TL;DR

This paper studies the asymptotic behavior of polynomials orthogonal on a cross-shaped contour with a Jacobi-type weight, revealing their limiting properties and potential applications in complex analysis.

Contribution

It provides new asymptotic results for non-Hermitian orthogonal polynomials on a cross with Jacobi-type weights, extending classical theory to complex contours.

Findings

Asymptotic formulas for orthogonal polynomials on a cross

Characterization of limiting zero distributions

Extension of classical orthogonal polynomial theory to complex contours

Abstract

We investigate asymptotic behavior of polynomials $ Q_n(z) $ satisfying non-Hermitian orthogonality relations $$ \int_\Delta s^kQ_n(s)\rho(s)\dd s =0, \quad k\in\{0,\ldots,n-1\}, $$ where $ \Delta := [-a,a]\cup [-\ic b,\ic b] $, $ a,b>0 $, and $ \rho(s) $ is a Jacobi-type weight.