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Zbl 0654.33004
Frenzen, C.L.; Wong, R.
Uniform asymptotic expansions of Laguerre polynomials. (English)
[J] SIAM J. Math. Anal. 19, No.5, 1232-1248 (1988). ISSN 0036-1410; ISSN 1095-7154/e

The Laguerre polynomials $L\sb n\sp{(\alpha)}(x)$ are considered for large n and fixed $\alpha >-1$. Two asymptotic expansions are obtained, valid uniformly in overlapping x-domains of the real axis. The approach is based on two integral representations, one leading to an expansion in terms of Airy functions, and the other one in terms of Bessel functions. Expansions of this type are derived earlier by Erdélyi, who only constructed leading terms by using the differential equations. Explicit remainders for the expansions are derived. The paper discusses in detail the conformal mappings needed for transforming the integrals to standard forms.
[N.M.Temme]

MSC 2000:: *33C45 Orthogonal polynomials and functions of hypergeometric type
41A60 Asymptotic problems in approximation

Keywords: Laguerre polynomials; Airy functions; Bessel functions

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