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A Uniform Asymptotic Expansion of the Jacobi Polynomials with Error Bounds

Published online by Cambridge University Press:  20 November 2018

C. L. Frenzen  and
R. Wong
C. L. Frenzen
Affiliation:
Southern Methodist University, Dallas, Texas
R. Wong
Affiliation:
Southern Methodist University, Dallas, Texas
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In a recent investigation of the asymptotic behavior of the Lebesgue constants for Jacobi polynomials, we encountered the problem of obtaining an asymptotic expansion for the Jacobi polynomials , as n → ∞, which is uniformly valid for θ in . The leading term of such an expansion is provided by the following well-known formula of “Hilb's type” [13, p. 197]:

(1.1)

where α > – 1, β real and ; c and are fixed positive numbers. Note that the remainder in (1.1) is always θ1/2O(n–3/2).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 5 , 01 October 1985 , pp. 979 - 1007
Copyright
Copyright © Canadian Mathematical Society 1985

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