Jacobi polynomials on the Bernstein ellipse

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.