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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 \(L_ n^{(\alpha)}(x)\). 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

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
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