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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Asymptotic behaviour of Jacobi polynomials and their zeros
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by Dimitar K. Dimitrov and Eliel J. C. dos Santos PDF
Proc. Amer. Math. Soc. 144 (2016), 535-545 Request permission

Abstract:

We obtain the explicit form of the expansion of the Jacobi polynomial $P_n^{(\alpha ,\beta )}(1-2x/\beta )$ in terms of the negative powers of $\beta$. It is known that the constant term in the expansion coincides with the Laguerre polynomial $L_n^{(\alpha )}(x)$. Therefore, the result in the present paper provides the higher terms of the asymptotic expansion as $\beta \rightarrow \infty$. The corresponding asymptotic relation between the zeros of Jacobi and Laguerre polynomials is also derived.
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Additional Information
  • Dimitar K. Dimitrov
  • Affiliation: Departamento de Matemática Aplicada, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil
  • MR Author ID: 308699
  • Email: dimitrov@ibilce.unesp.br
  • Eliel J. C. dos Santos
  • Affiliation: Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Campinas-SP, 13083-970, Brazil
  • MR Author ID: 1138359
  • Email: ra115031@ime.unicamp.br
  • Received by editor(s): May 11, 2014
  • Received by editor(s) in revised form: October 10, 2014, and November 4, 2014
  • Published electronically: October 1, 2015
  • Additional Notes: The authors’ research was supported by the Brazilian foundations CNPq under Grant 307183/2013–0 and FAPESP under Grant 2009/13832–9.
  • Communicated by: Walter Van Assche
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 535-545
  • MSC (2010): Primary 26C10, 33C45
  • DOI: https://doi.org/10.1090/proc/12689
  • MathSciNet review: 3430832