Asymptotic computation of classical orthogonal polynomials

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 $n$.

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