### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 073, 9 pages      arXiv:1804.06749      https://doi.org/10.3842/SIGMA.2018.073
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

### Asymptotic Expansions of Jacobi Polynomials for Large Values of $\beta$ and of Their Zeros

Amparo Gil a, Javier Segura a and Nico M. Temme c
a) Departamento de Matemática Aplicada y CC, de la Computación, ETSI Caminos, Universidad de Cantabria, 39005-Santander, Spain
b) Departamento de Matemáticas, Estadistica y Computación, Universidad de Cantabria, 39005 Santander, Spain
c) IAA, 1825 BD 25, Alkmaar, The Netherlands
Former address: Centrum Wiskunde & Informatica (CWI), Science Park 123, 1098 XG Amsterdam, The Netherlands

Received April 19, 2018, in final form July 12, 2018; Published online July 17, 2018

Abstract
Asymptotic approximations of Jacobi polynomials are given for large values of the $\beta$-parameter and of their zeros. The expansions are given in terms of Laguerre polynomials and of their zeros. The levels of accuracy of the approximations are verified by numerical examples.

Key words: Jacobi polynomial; large-beta asymptotics; Laguerre polynomial.

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References

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3. Gil A., Segura J., Temme N.M., Non-iterative computation of Gauss-Jacobi quadrature by asymptotic expansions for large degree, arXiv:1804.07076.
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