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Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions

Abstract

Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials \(L_{n}^{(\alpha )}(x)\), as well as complementary confluent hypergeometric functions. The expansions are valid for n large and α small or large, uniformly for unbounded real and complex values of x. The new expansions extend the range of computability of \(L_{n}^{(\alpha )}(x)\) compared to previous expansions, in particular with respect to higher terms and large values of α. Numerical evidence of their accuracy for real and complex values of x is provided.

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Acknowledgements

The authors acknowledge support from Ministerio de Economía y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE).

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Correspondence to T. M. Dunster.

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Communicated by: Robert Schaback

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Dunster, T.M., Gil, A. & Segura, J. Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions. Adv Comput Math 44, 1441–1474 (2018). https://doi.org/10.1007/s10444-018-9589-5

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  • DOI: https://doi.org/10.1007/s10444-018-9589-5

Keywords

  • Asymptotic expansions
  • Laguerre polynomials
  • Confluent hypergeometric functions
  • Turning point theory
  • WKB methods
  • Numerical computation

Mathematics Subject Classification (2010)

  • 34E05
  • 33C45
  • 33C15
  • 34E20
  • 33F05