On the Relative Extrema of the Jacobi Polynomials $P_n^{^}{(0, - 1)} (x)$

Asymptotic approximations, complete with error bounds, are constructed for the Jacobi polynomials $P_{n - 1}^{^}{(0,1)} (\cos \theta )$ and $P_{n - 1}^{^}{(1,0)} (\cos \theta )$, as $n \to \infty $, which hold uniformly with respect to $\theta \epsilon [0,0.78\pi ]$. A corresponding approximation is also obtained for the zeros $\theta _{k,n} $ of $P_{n - 1}^{^}{(1,0)} (\cos \theta )$. These results are then used to prove the following conjecture of Askey: If $\nu _{k,n} $ denotes the relative extreme of the Jacobi polynomial $P_n^{^}{(0, - 1)} (x)$, ordered so that $\nu _{k + 1,n} $ lies to the left of $\nu _{k,n} $, then $| {\nu _{k,n - 1} } | < | {\nu _{k,n} } |$ for $k = 1, \ldots ,n - 1$ and $n = 1,2, \ldots $.