# Jacobi polynomials on the Bernstein ellipse

## TL;DR

This paper derives explicit formulas and asymptotic estimates for Jacobi polynomials on the Bernstein ellipse, revealing where their maxima and minima occur, which advances understanding of their convergence in spectral interpolation.

## Contribution

It provides new explicit representations and asymptotic estimates for Jacobi polynomials on the Bernstein ellipse, extending known results to broader parameter ranges.

## Key findings

- Maximum of |P_n^{(α,β)}(z)| occurs at endpoints of the major axis if α+β ≥ -1.
- Minimum value for Gegenbauer polynomials occurs at endpoints of the minor axis.
- Extended previous results to more general cases of Jacobi polynomials.

## Abstract

In this paper, we are concerned with Jacobi polynomials $P_n^{(\alpha,\beta)}(x)$ on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of $P_n^{(\alpha,\beta)}(x)$ is derived in the variable of parametrization. This formula further allows us to show that the maximum value of $\left|P_n^{(\alpha,\beta)}(z)\right|$ over the Bernstein ellipse is attained at one of the endpoints of the major axis if $\alpha+\beta\geq -1$. For the minimum value, we are able to show that for a large class of Gegenbauer polynomials (i.e., $\alpha=\beta$), it is attained at two endpoints of the minor axis. These results particularly extend those previously known only for some special cases. Moreover, we obtain a more refined asymptotic estimate for Jacobi polynomials on the Bernstein ellipse.