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Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

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Abstract

In this paper, the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|2αe−(x4+tx2), x ∈ R, where α is a constant larger than −1/2 and t is any real number. They consider this problem in three separate cases: (i) c > −2, (ii) c = −2, and (iii) c < −2, where c:= tN−1/2 is a constant, N = n + α and n is the degree of the polynomial. In the first two cases, the support of the associated equilibrium measure μt is a single interval, whereas in the third case the support of μt consists of two intervals. In each case, globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993).

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Acknowledgements

The first author is grateful to Dan Dai, Yu Lin, Xiang-Sheng Wang and Lun Zhang for useful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, China

    Zhi-Tao Wen

  2. Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Hong Kong, China

    Zhi-Tao Wen & Roderick Wong

  3. Institut Franco-Chinois de l’Energie Nucléaire, Sun Yat-sen University, Guangzhou, 510275, China

    Shuai-Xia Xu

Authors
  1. Zhi-Tao Wen
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  2. Roderick Wong
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  3. Shuai-Xia Xu
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Corresponding author

Correspondence to Zhi-Tao Wen.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11771090, 11571376).

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Wen, ZT., Wong, R. & Xu, SX. Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight. Chin. Ann. Math. Ser. B 39, 553–596 (2018). https://doi.org/10.1007/s11401-018-0082-8

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  • DOI: https://doi.org/10.1007/s11401-018-0082-8

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