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Asymptotic Monotonicity of the Relative Extrema of Jacobi Polynomials
Published online by Cambridge University Press: 20 November 2018
Abstract
If μk,n(α,β) denotes the relative extrema of the Jacobi polynomial P(α,β)n(x), ordered so that μk+1,n(α,β) lies to the left of μk,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 μk,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.
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- Copyright © Canadian Mathematical Society 1994
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