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Asymptotic estimates for Laguerre polynomials. (English) Zbl 0688.33007
A summary is given of recent results concerning the asymptotic behaviour of the Laguerre polynomials ${L}_{n}^{\left(\alpha \right)}\left(x\right)$. First the results are summarized of a paper of Frenzen and Wong in which $n\to \infty$ and $\alpha >-1$ is fixed. Two different expansions are needed in that case, one with a J-Bessel function and one with an Airy function as main approximant. Second, three other forms are given in which $\alpha$ is not necessarily fixed. Again Bessel and Airy functions are used, and in another form the comparison function is a Hermite polynomial. A numerical verification of the new expansion in terms of the Hermite polynomial is given by comparing the zeroes of the approximant with the related zeros of the Laguerre polynomial.
Reviewer: N.M.Temme
##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
##### Keywords:
Laguerre polynomial
##### References:
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