Numerische Mathematik

, Volume 116, Issue 2, pp 269–289 | Cite as

Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions



Expansions in terms of Bessel functions are considered of the Kummer function 1F1(a; c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the expansions is explained for large values of the parameter a of the Kummer function.


Kummer functions Confluent hypergeometric functions Bessel functions Asymptotic expansions Computation of special functions 

Mathematics Subject Classification (2000)

33C10 33C15 41A30 41A60 65D20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abad J., Sesma J.: Buchholz polynomials: a family of polynomials relating solutions of confluent hypergeometric and Bessel equations. J. Comput. Appl. Math. 101(1–2), 237–241 (1999)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abad J., Sesma J.: A new expansion of the confluent hypergeometric function in terms of modified Bessel functions. J. Comput. Appl. Math. 78(1), 97–101 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol 55 of National Bureau of Standards Applied Mathematics Series. U.S. Government Printing Office, Washington (1964)Google Scholar
  4. 4.
    Buchholz, H.: The Confluent Hypergeometric Function with Special Emphasis on Its Applications. Translated from the German by Lichtblau, H., Wetzel, K. Springer Tracts in Natural Philosophy, vol. 15. Springer, New York (1969)Google Scholar
  5. 5.
    Chiccoli C., Lorenzutta S., Maino G.: A note on a Tricomi expansion for the generalized exponential integral and related functions. Nuovo Cimento B (11) 103(5), 563–568 (1989)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gil A., Segura J., Temme N.M.: Numerical Methods for Special Functions. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2007)MATHGoogle Scholar
  7. 7.
    Kreuser, P.: Über das Verhalten der Integrale homogener linearer Differenzengleichungen im Unendlichen. PhD thesis, Diss. Tübingen, 48S (1914)Google Scholar
  8. 8.
    López J.L., Temme N.M.: Two-point Taylor expansions of analytic functions. Stud. Appl. Math. 109(4), 297–311 (2002)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    López, J.L., Temme, N.M.: Multi-point Taylor expansions of analytic functions. Trans. Amer. Math. Soc. 356(11), 4323–4342 (electronic) (2004)Google Scholar
  10. 10.
    López J.L., Temme N.M.: Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363(1), 197–208 (2010)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Maino G., Menapace E., Ventura A.: Computation of parabolic cylinder functions by means of a Tricomi expansion. J. Comput. Phys. 40(2), 294–304 (1981)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Olver, F.W.J.: Asymptotics and Special Functions. AKP Classics. A K Peters Ltd., Wellesley (1997). Reprint, with corrections, of original Academic Press edition (1974)Google Scholar
  13. 13.
    Slater L.J.: Confluent Hypergeometric Functions. Cambridge University Press, New York (1960)MATHGoogle Scholar
  14. 14.
    Temme N.M.: Special Functions. An Introduction to the Classical Functions of Mathematical Physics. A Wiley-Interscience Publication. Wiley, New York (1996)Google Scholar
  15. 15.
    Tricomi F.: Sulle funzioni ipergeometriche confluenti. Ann. Mat. Pura Appl. (4) 26, 141–175 (1947)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Wong, R.: Asymptotic Approximations of Integrals, vol. 34 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Corrected reprint of the 1989 original (2001)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Departamento de Ingeniería Matemática e InformáticaUniversidad Pública de NavarraPamplonaSpain
  2. 2.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands

Personalised recommendations