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Uniform asymptotic expansions of Laguerre polynomials. (English) Zbl 0654.33004
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.
Reviewer: N.M.Temme

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
##### Keywords:
Laguerre polynomials; Airy functions; Bessel functions
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