A uniform asymptotic expansion for Jacobi polynomials via uniform treatment of Darboux's method

Authors:
Xiao-Xi Bai;Yu-Qiu Zhao
Affiliations:
Department of Mathematics , ZhongShan University, GuangZhou 510275, PR China;Department of Mathematics , ZhongShan University, GuangZhou 510275, PR China
Venue:
Journal of Approximation Theory
Year:
2007

Citing 1
Cited 1

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Full length article: Polynomial Schauder basis of optimal degree with Jacobi orthogonality

Journal of Approximation Theory

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Abstract

Uniform asymptotic properties of the classical Jacobi polynomials have been studied via various approaches other than Darboux's method. In this note, by using ideas of the uniform treatment of Darboux's method, an asymptotic expansion, in terms of the Bessel function of the first kind and its derivative, is obtained for Jacobi polynomials P"n^(^@a^,^@b^)(cos@q). The expansion is uniformly valid for @q@?[0,@p-@e], @e being an arbitrary positive constant.