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Error bounds for the asymptotic expansions of the Hermite polynomials

Part of: Approximations and expansions

Published online by Cambridge University Press:  26 January 2022

Wei Shi ,
Gergő Nemes ,
Xiang-Sheng Wang Open the ORCID record for Xiang-Sheng Wang [Opens in a new window]  and
Roderick Wong
Wei Shi
Affiliation:
College of Science, Huazhong Agricultural University, Wuhan 430070, P. R. China
Gergő Nemes
Affiliation:
Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, Budapest H-1053, Hungary
Xiang-Sheng Wang
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70503, USA ([email protected])
Roderick Wong
Affiliation:
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
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Abstract

In this paper, we present explicit and computable error bounds for the asymptotic expansions of the Hermite polynomials with Plancherel–Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory interval, or it is the turning point, are considered separately. We introduce the ‘branch cut’ technique to express the error terms as integrals on the contour taken as the one-sided limit of curves approaching the branch cut. This new technique enables us to derive simple error bounds in terms of elementary functions. We also provide recursive procedures for the computation of the coefficients appearing in the asymptotic expansions.

Keywords

Error boundsasymptotic expansionsHermite polynomials

MSC classification

Primary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 2 , April 2023 , pp. 417 - 440
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Bennett, T., Howls, C. J., Nemes, G. and Olde Daalhuis, A. B.. Globally exact asymptotics for integrals with arbitrary order saddles. SIAM J. Math. Anal. 50 (2018), 2144–2177.CrossRefGoogle Scholar
Berry, M. V. and Howls, C. J.. Hyperasymptotics for integrals with saddles. Proc. Roy. Soc. London Ser. A 434 (1991), 657–675.Google Scholar
Boyd, W. G. C.. Error bounds for the method of steepest descents. Proc. Roy. Soc. London Ser. A 440 (1993), 493–518.Google Scholar
Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. and Zhou, X.. Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 (1999), 1491–1552.3.0.CO;2-#>CrossRefGoogle Scholar
Deift, P. and Zhou, X.. A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137 (1993), 295–368.Google Scholar
Lauwerier, H. A.. The calculation of the coefficients of certain asymptotic series by means of linear recurrent relations. Appl. Sci. Res. B 2 (1952), 77–84.CrossRefGoogle Scholar
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.1.3 of 2021-09-15, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds..Google Scholar
Olver, F. W. J.. Asymptotics and Special Functions (Academic Press, New York, 1974).Google Scholar
Olver, F. W. J.. Asymptotic approximations and error bounds. SIAM Rev. 22 (1980), 188–203.CrossRefGoogle Scholar
van Veen, S. C.. Asymptotische Entwicklung und Nullstellenabschätzung der Hermiteschen Funktionen. Math. Annal. 105 (1931), 408–436.CrossRefGoogle Scholar
Wong, R.. Error bounds for asymptotic expansions of integrals. SIAM Rev. 22 (1980), 401–435.CrossRefGoogle Scholar
Wong, R.. Asymptotic Approximations of Integrals (Academic Press, Boston, 1989).Google Scholar
Wong, R.. Asymptotics of linear recurrences. Anal. Appl. 12 (2014), 463–484.CrossRefGoogle Scholar
Wong, R. and Zhao, Y.-Q.. On a uniform treatment of Darboux's method. Constr. Approx. 21 (2005), 225–255.CrossRefGoogle Scholar