On the Relative Extrema of the Jacobi Polynomials $P_n^{(0, - 1)} (x)$

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Author(s)

  • Sue Cheun Roderick WONG
  • J. M. Zhang

Detail(s)

Original languageEnglish
Pages (from-to)776-811
Journal / PublicationSIAM Journal on Mathematical Analysis
Volume25
Issue number2
Publication statusPublished - 1994
Externally publishedYes

Abstract

Asymptotic approximations, complete with error bounds, are constructed for the Jacobi polynomials $P_{n - 1}^{(0,1)} (\cos \theta )$ and $P_{n - 1}^{(1,0)} (\cos \theta )$, as $n \to \infty $, which hold uniformly with respect to $\theta \epsilon [0,0.78\pi ]$. A corresponding approximation is also obtained for the zeros $\theta _{k,n} $ of $P_{n - 1}^{(1,0)} (\cos \theta )$. These results are then used to prove the following conjecture of Askey: If $\nu _{k,n} $ denotes the relative extreme of the Jacobi polynomial $P_n^{(0, - 1)} (x)$, ordered so that $\nu _{k + 1,n} $ lies to the left of $\nu _{k,n} $, then $| {\nu _{k,n - 1} } | < | {\nu _{k,n} } |$ for $k = 1, \ldots ,n - 1$ and $n = 1,2, \ldots $.

Citation Format(s)

On the Relative Extrema of the Jacobi Polynomials $P_n^{(0, - 1)} (x)$. / WONG, Sue Cheun Roderick; Zhang, J. M.
In: SIAM Journal on Mathematical Analysis, Vol. 25, No. 2, 1994, p. 776-811.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review