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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] Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
  • [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