Asymptotic behaviour of Jacobi polynomials and their zeros

Dimitrov, Dimitar; dos Santos, Eliel

doi:10.1090/proc/12689

Asymptotic behaviour of Jacobi polynomials and their zeros
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by Dimitar K. Dimitrov and Eliel J. C. dos Santos

Proc. Amer. Math. Soc. 144 (2016), 535-545

DOI: https://doi.org/10.1090/proc/12689

Published electronically: October 1, 2015

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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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