# Gegenbauer and other planar orthogonal polynomials on an ellipse in the complex plane

## TL;DR

This paper demonstrates that classical orthogonal polynomials like Gegenbauer and Jacobi are also orthogonal within an elliptical region in the complex plane, expanding their applications and connecting to random matrix theory.

## Contribution

It extends the orthogonality of classical polynomials to elliptical regions in the complex plane and explores their connections to Bergman spaces and random matrix ensembles.

## Key findings

- Orthogonality of Gegenbauer and Jacobi polynomials on an ellipse in the complex plane.
- Connection to weighted Bergman spaces and Selberg integrals.
- Limit cases recover known polynomials and their properties.

## Abstract

We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polynomials $C_n^{(1+\alpha)}(z)$ for $\alpha>-1$ containing the Legendre polynomials $P_n(z)$, and the subset $P_n^{(\alpha+\frac12,\pm\frac12)}(z)$ of the Jacobi polynomials. These polynomials provide an orthonormal basis and the corresponding weighted Bergman space forms a complete metric space. This leads to a certain family of Selberg integrals in the complex plane. We recover the known orthogonality of Chebyshev polynomials of first up to fourth kind. The limit $\alpha\to\infty$ leads back to the known Hermite polynomials orthogonal in the entire complex plane. When the ellipse degenerates to a circle we obtain the weight function and monomials known from the determinantal point process of the ensemble of truncated unitary random matrices.