Jacobi Polynomials on the Bernstein Ellipse

In this paper, we are concerned with Jacobi polynomials P n ( α , β ) ( x ) on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of P n ( α , β ) ( x ) is derived in the variable of parametrization. This formula further allows us to show that the maximum value of P n ( α , β ) ( z ) over the Bernstein ellipse is attained at one of the endpoints of the major axis if α + β ≥ - 1 . For the minimum value, we are able to show that for a large class of Gegenbauer polynomials (i.e., α = β ), 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.