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Asymptotic formulas for ultraspherical polynomials $ P\sb n\sp \lambda(x)$ and their zeros for large values of $ \lambda$


Authors: Árpád Elbert and Andrea Laforgia
Journal: Proc. Amer. Math. Soc. 114 (1992), 371-377
MSC: Primary 33C55; Secondary 33C45
DOI: https://doi.org/10.1090/S0002-9939-1992-1089404-X
MathSciNet review: 1089404
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Abstract: For $ \lambda > - 1/2$ we denote by $ P_n^{(\lambda )}(x)$ the ultraspherical polynomial of degree $ n$ and by $ x_{n,k}^{(\lambda )}$ and $ {h_{n,k}}(k = 1,2, \ldots ,n)$ the $ k$th zeros of $ P_n^{(\lambda )}(x)$ and of the Hermite polynomial $ {H_n}(x)$, respectively. In this paper we establish the following formulas

$\displaystyle {\lambda ^{ - n/2}}P_n^{(\lambda )}\left( {\frac{x}{{\sqrt \lambd... ...ts_{j = 0}^{n - 1} {{\lambda ^{ - j}}{Q_{nj}}(x)\,{\text{for}}\,\lambda \ne 0} $

and

$\displaystyle x_{n,k}^{(\lambda )} = {h_{n,k}}{\lambda ^{ - 1/2}} - \frac{{{h_{... ...n,k}^4} \right){\lambda ^{ - 5/2}} + O({\lambda ^{ - 7/2}}),\lambda \to \infty $

where $ {Q_{n0}}(x) = {H_n}(x)/n!$ and $ {Q_{nj}}(x)(j = 1,2, \ldots ,n - 1)$ are polynomials specified in Theorem 1. Finally we show that the positive (negative) zeros of $ P_n^{(\lambda )}(x)$ are convex (concave) functions of $ \lambda $, provided $ \lambda $ is sufficiently large.

References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I. A. Stegun, eds., Handbook of mathematical functions, Applied Math. Ser. 55, National Bureau of Standards, Washington, D. C., 1964. MR 0167642 (29:4914)
  • [2] G. Szegö, Orthogonal polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. vol. 23, Amer. Math. Soc., Providence, RI, 1975.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1089404-X
Keywords: Zeros of ultraspherical polynomials, asymptotic expansion
Article copyright: © Copyright 1992 American Mathematical Society

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