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Asymptotic behaviour of Jacobi polynomials and their zeros


Authors: Dimitar K. Dimitrov and Eliel J. C. dos Santos
Journal: Proc. Amer. Math. Soc. 144 (2016), 535-545
MSC (2010): Primary 26C10, 33C45
DOI: https://doi.org/10.1090/proc/12689
Published electronically: October 1, 2015
MathSciNet review: 3430832
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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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Dimitar K. Dimitrov
Affiliation: Departamento de Matemática Aplicada, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil
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
Email: ra115031@ime.unicamp.br

DOI: https://doi.org/10.1090/proc/12689
Keywords: Jacobi polynomials, Laguerre polynomials, zeros, asymptotics
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
Article copyright: © Copyright 2015 American Mathematical Society