Let denote the -th Jacobi polynomial and let be the abscissas of the relative extrema of ordered by . Set
A remarkable result obtained in the paper is the asymptotic representation, as and for each fixed ,
where and is the -th positive zero of the Bessel function . This representation, which corrects an earlier result of R. Cooper [Proc. Cambridge Phil. Soc. 46, 549- 554 (1950; Zbl 0038.223)], is not sufficient to prove a monotonicity property of the extrema conjectured by Askey, i.e. that for ,
The authors are able to overcome this difficulty and show that Askey’s conjecture is true at least in the asymptotic sense. This is done by using another more powerful representation of , derived from a uniform asymptotic approximation of the Jacobi polynomial.