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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the zeros of Jacobi polynomials $ P\sb{n}\sp{(\alpha \sb{n},\beta \sb{n})}(x)$


Authors: D. S. Moak, E. B. Saff and R. S. Varga
Journal: Trans. Amer. Math. Soc. 249 (1979), 159-162
MSC: Primary 33A65
MathSciNet review: 526315
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Abstract: If $ {r_n}$ and $ {s_n}$ denote, respectively, the smallest and largest zeros of the Jacobi polynomial $ P_n^{({\alpha _n},{\beta _n})}$, where $ {\alpha _n}\, > \,1$, $ {\beta _n}\, - \,1$, and if $ {\lim _{n \to \infty }}\,{\alpha _n}/(2n\, + \,{\alpha _n}\, + \,{\beta _n}\, + \,1)\, = \,a $ and if $ {\lim _{n \to \infty }}{\beta _n}/(2n\, + \,{\alpha _n}\, + \,{\beta _n}\, + \,1)\, = \,b $, then the numbers $ {r_{a,b}}$ and $ {s_{a,b}}$ are determined where

$\displaystyle \mathop {\lim }\limits_{n \to \infty } \,{r_{n\,}}\, = \,{r_{a,b}},\mathop {\lim }\limits_{n \to \infty } \,{s_{n\,}}\, = \,{s_{a,b}}$

. Furthermore, the zeros of $ \{ P_n^{({\alpha _n},{\beta _n})}(x)\} _{n = 0}^\infty $ are dense in $ [{r_{a,b}},{s_{a,b}}]$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0526315-8
PII: S 0002-9947(1979)0526315-8
Keywords: Jacobi polynomials, Sturm Comparison Theorem
Article copyright: © Copyright 1979 American Mathematical Society