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Global asymptotic expansions of the Laguerre polynomials—a Riemann–Hilbert approach

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Abstract

By using the steepest descent method for Riemann–Hilbert problems introduced by Deift–Zhou (Ann Math 137:295–370, 1993), we derive two asymptotic expansions for the scaled Laguerre polynomial \(L^{(\alpha)}_n(\nu z)\) as n→∞, where ν=4n+2α+2. One expansion holds uniformly in a right half-plane \(\text{Re}\; z\geq \delta_1, 0<\delta_1<1\), which contains the critical point z=1; the other expansion holds uniformly in a left half-plane \(\text{Re}\; z\leq 1-\delta_2, 0<\delta_2<1-\delta_1\), which contains the other critical point z=0. The two half-planes together cover the entire complex z-plane. The critical points z=1 and z=0 correspond, respectively, to the turning point and the singularity of the differential equation satisfied by \(L^{(\alpha)}_n(\nu z)\).

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Authors and Affiliations

  1. Department of Mathematics, Fudan University, Shanghai, China

    W.-Y. Qiu

  2. Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

    R. Wong

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  1. W.-Y. Qiu
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  2. R. Wong
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Correspondence to R. Wong.

Additional information

In memory of our good friend Professor Luigi Gatteschi.

This author is supported partially by Chinese NNSF No.10571028.

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Qiu, WY., Wong, R. Global asymptotic expansions of the Laguerre polynomials—a Riemann–Hilbert approach. Numer Algor 49, 331–372 (2008). https://doi.org/10.1007/s11075-008-9159-x

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