Asymptotic Monotonicity of the Relative Extrema of Jacobi Polynomials

If μk,n (α,β) denotes the relative extrema of the Jacobi polynomial P(α,β) n(x), ordered so that μ k+1,n (α,β) lies to the left of μ k,n (α,β), then R. A. Askey has conjectured twenty years ago that for for k = 1,…, n — 1 and n = 1,2,=. In this paper, we give an asymptotic expansion for μ k,n (α,β) when k is fixed and n → ∞, which corrects an earlier result of R. Cooper (1950). Furthermore, we show that Askey's conjecture is true at least in the asymptotic sense.