iBet uBet web content aggregator. Adding the entire web to your favor.
iBet uBet web content aggregator. Adding the entire web to your favor.



Link to original content: https://unpaywall.org/10.1007/S10444-018-9589-5
Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions | Advances in Computational Mathematics Skip to main content
Log in

Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Atia, M.J., Martínez-Finkelshtein, A., Martínez-González, P., Thabet, F.: Quadratic differentials and asymptotics of laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416(1), 52–80 (2014). https://doi.org/10.1016/j.jmaa.2014.02.040

    Article  MathSciNet  MATH  Google Scholar 

  2. Borwein, D., Borwein, J.M., Crandall, R.E.: Effective Laguerre asymptotics. SIAM J. Numer. Anal. 46(6), 3285–3312 (2008). https://doi.org/10.1137/07068031X

    Article  MathSciNet  MATH  Google Scholar 

  3. Dai, D., Ismail, M.E.H., Wang, J.: Asymptotics for Laguerre polynomials with large order and parameters. J. Approx. Theory. 193, 4–19 (2015). https://doi.org/10.1016/j.jat.2014.03.009

    Article  MathSciNet  MATH  Google Scholar 

  4. Dai, D., Qiu, W., Wang, J.: Uniform asymptotics for orthogonal polynomials with exponential weight on the positive real axis. Asymptot. Anal. 89(1-2), 1–19 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Deaño, A., Huertas, E.J., Marcellán, F.: Strong and ratio asymptotics for Laguerre polynomials revisited. J. Math. Anal. Appl. 403(2), 477–486 (2013). https://doi.org/10.1016/j.jmaa.2013.02.039

    Article  MathSciNet  MATH  Google Scholar 

  6. Dunster, T.M.: Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20(3), 744–760 (1989). https://doi.org/10.1137/0520052

    Article  MathSciNet  MATH  Google Scholar 

  7. Dunster, T.M.: Asymptotics of the eigenvalues of the rotating harmonic oscillator. J. Comput. Appl. Math. 93(1), 45–73 (1998). https://doi.org/10.1016/S0377-0427(98)00070-3

    Article  MathSciNet  MATH  Google Scholar 

  8. Dunster, T.M., Gil, A., Segura, J.: Computation of asymptotic expansions of turning point problems via Cauchy’s integral formula: Bessel functions. Constr. Approx. 46(3), 645–675 (2017). https://doi.org/10.1007/s00365-017-9372-8

    Article  MathSciNet  MATH  Google Scholar 

  9. Farid Khwaja, S., Olde Daalhuis, A.B.: Computation of the coefficients appearing in the uniform asymptotic expansions of integrals. Stud. Appl. Math. (2017). https://doi.org/10.1111/sapm.12172

    Article  MathSciNet  Google Scholar 

  10. Frenzen, C.L., Wong, R.: Uniform asymptotic expansions of Laguerre polynomials. SIAM J. Math. Anal. 19(5), 1232–1248 (1988). https://doi.org/10.1137/0519087

    Article  MathSciNet  MATH  Google Scholar 

  11. Gil, A., Segura, J., Temme, N.M.: Efficient computation of Laguerre polynomials. Comput. Phys. Commun. 210, 124–131 (2017). https://doi.org/10.1016/j.cpc.2016.09.002

    Article  MathSciNet  MATH  Google Scholar 

  12. Huybrechs, D., Opsomer, P.: Construction and implementation of asymptotic expansions for laguerre- type orthogonal polynomials. IMA J. Numer. Anal. (to appear). https://doi.org/10.1093/imanum/drx030

    Article  MathSciNet  Google Scholar 

  13. Olde Daalhuis, A.B.: Confluent hypergeometric functions. In: NIST Handbook of Mathematical Functions, pp. 321–349. U.S. Department Commerce, Washington, DC (2010)

  14. Olver, F.W.J.: Whittaker functions with both parameters large: uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh. Sect. A 86(3-4), 213–234 (1980). https://doi.org/10.1017/S0308210500012130

    Article  MathSciNet  MATH  Google Scholar 

  15. Olver, F.W.J.: Asymptotics and Special Functions. AKP Classics. A K Peters Ltd., Wellesley (1997). Reprint of the 1974 original [Academic Press, New York]

    Book  Google Scholar 

  16. Olver, F.W.J.: Airy and Related Functions. In: NIST Handbook of Mathematical Functions, pp. 193–213. U.S. Dept. Commerce, Washington, DC (2010)

  17. Qiu, W.Y., Wong, R.: Global asymptotic expansions of the Laguerre polynomials—a Riemann-Hilbert approach. Numer. Algorithms 49(1-4), 331–372 (2008). https://doi.org/10.1007/s11075-008-9159-x

    Article  MathSciNet  MATH  Google Scholar 

  18. Temme, N.M.: Asymptotic estimates for Laguerre polynomials. Z. Angew. Math. Phys. 41(1), 114–126 (1990). https://doi.org/10.1007/BF00946078

    Article  MathSciNet  MATH  Google Scholar 

  19. Temme, N.M.: Asymptotic Methods for Integrals. Series in Analysis, vol. 6. World Scientific Publishing Co. Pte. Ltd, Hackensack (2015)

    Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to T. M. Dunster.

Additional information

Communicated by: Robert Schaback

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

Mathematics Subject Classification (2010)

Navigation