The classical orthogonal polynomials (Hermite, Laguerre and Jacobi) are involved in a vast number of applications in physics and engineering. When large degrees n are needed, the use of recursion to compute the polynomials is not a good strategy for computation and a more efficient approach, such as the use of asymptotic expansions,is recommended. In this paper, we give an overview of the asymptotic expansions considered in [8] for computing Laguerre polynomials L(α)n(x) for bounded values of the parameter α. Additionally, we show examples of the computational performance of an asymptotic expansion for L(α)n(x) valid for large values of α and n. This expansion was used in [6] as starting point for obtaining asymptotic approximations to the zeros. Finally, we analyze the expansions considered in [9], [10] and [11] to compute the Jacobi polynomials for large degrees n.

doi.org/10.1007/978-3-030-56190-1
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7th Ibero-American Workshop in Orthogonal Polynomials and Applications

Gil, A, Segura, J, & Temme, N.M. (2018). Asymptotic computation of classical orthogonal polynomials. In Orthogonal Polynomials: Current Trends and Applications. Proceedings of the 7th EIBPOA Conference. doi:10.1007/978-3-030-56190-1