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Zbl 0805.33014
Wong, R.; Zhang, J.-M.
On the relative extrema of the Jacobi polynomials $P\sb 0\sp{(0,-1)}(x)$. (English)
[J] SIAM J. Math. Anal. 25, No.2, 776-811 (1994). ISSN 0036-1410; ISSN 1095-7154/e

Let $\nu\sb{k,n}$, $k= 1,\dots,n$; $n= 1,2,\dots$ be the successive relative extrema of $P\sp{(0,-1)}\sb n(x)/P\sp{(0,-1)}\sb n(1)$ when $x$ decreases from $+1$ to $-1$. With a view to proving the conjecture of Askey that $\vert\nu\sb{k,n}\vert< \vert\nu\sb{k,n+1}\vert$, first the authors derive some asymptotic approximations for the Jacobi polynomials $P\sp{(0,1)}\sb{n-1}(\cos\theta)$ and $P\sp{(1,0)}\sb{n-1}(\cos\theta)$, as $n$ tends to $\infty$, which lead them to obtain corresponding approximations for the zeros of $P\sp{(1,0)}\sb{n-1}(\cos\theta)$. These asymptotic approximations are then used to prove the conjecture stated above.
[J.P.Singhal (Baroda)]

MSC 2000:: *33C45 Orthogonal polynomials and functions of hypergeometric type
41A60 Asymptotic problems in approximation

Keywords: zeros; relative extrema; uniform; Jacobi polynomials

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