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Asymptotic Monotonicity of the Relative Extrema of Jacobi Polynomials

Published online by Cambridge University Press:  20 November 2018

R. Wong  and
J.-M. Zhang
R. Wong
Affiliation:
Department of Applied Mathematics, University of Manitoba Winnipeg, ManitobaR3T 2N2
J.-M. Zhang
Affiliation:
Department of Applied Mathematics, Tsinghua University, Beijing China
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Abstract

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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.

Keywords

33C4541A60Jacobi polynomialszerosrelative extremauniform asymptotic approximation
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 6 , 01 December 1994 , pp. 1318 - 1337
Copyright
Copyright © Canadian Mathematical Society 1994

References

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