Numerical Algorithms

, Volume 76, Issue 2, pp 377–391 | Cite as

New properties of a certain method of summation of generalized hypergeometric series

  • Rafał Nowak
  • Paweł Woźny
Original Paper


In a recent paper (Appl. Math. Comput. 215:1622–1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the convergence acceleration theorem in the case of summation of generalized hypergeometric series. Some new theoretical results and illustrative numerical examples are given.


Convergence acceleration Summation Generalized hypergeometric series 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 10th Printing Edn. National Bureau of Standards, Washington (1972)MATHGoogle Scholar
  2. 2.
    Aitken, A.C.: On Bernoulli’s numerical solution of algebraic equations. Proc. Roy. Soc. Edinburgh 46, 289–305 (1926)CrossRefMATHGoogle Scholar
  3. 3.
    Andrews, G.E., Askey, R., Roy, R.: Special functions. Cambridge University Press, Cambridge (1999)CrossRefMATHGoogle Scholar
  4. 4.
    Baker, Jr, G.A.: Essentials of Padé approximants. Academic Press, New York (1975)MATHGoogle Scholar
  5. 5.
    Baker Jr, G.A., Graves-Morris, P.: Padé approximants. Part I: basic theory. II: Extensions and applications. Addison-Wesley, Reading (1981)MATHGoogle Scholar
  6. 6.
    Brezinski, C.: Convergence acceleration during the 20th century. J. Comput. Appl. Math. 122(1–2), 1–21 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brezinski, C., Redivo Zaglia, M.: Extrapolation methods: theory and practice, studies in computational mathematics, vol. 2, North-Holland (1991)Google Scholar
  8. 8.
    Ċízek, J., Zamastil, J., Skála, L.: New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field. J. Math. Phys. 44(3), 962–968 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Clark, W.D., Gray, H.L., Adams, J.E.: A note on the T-transformation of Lubkin. J. Res. Natl. Bur. Stand. 73B, 25–29 (1969)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Homeier, H.H.H.: A hierarchically consistent, iterative sequence transformation. Numer. Algorithms 8(1), 47–81 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Homeier, H.H.H.: Scalar Levin-type sequence transformations. In: Brezinski, C. (ed.) Numerical Analysis 2000, Vol. 2: Interpolation and Extrapolation, pp. 81–147 (2000)Google Scholar
  12. 12.
    Kim, Y.S., Rathie, A.K., Paris, R.B.: On two Thomae-type transformations for hypergeometric series with integral parameter differences. Math. Commun. 19(1), 111–118 (2014)MathSciNetMATHGoogle Scholar
  13. 13.
    Levin, D.: Development of non-linear transformations for improving convergence of sequences. J. Comput. Math. 3, 371–388 (1973)MathSciNetMATHGoogle Scholar
  14. 14.
    Lewanowicz, S., Paszkowski, S.: An analytic method for convergence acceleration of certain hypergeometric series. Math. Comput. 64(210), 691–713 (1995)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and theorems for the special functions of mathematical physics. Springer, New York (1966)CrossRefMATHGoogle Scholar
  16. 16.
    Miller, A.R., Paris, R.B.: Certain transformations and summations for generalized hypergeometric series with integral parameter differences. Integr. Transf. Spec. F. 22(1–3), 67–77 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Miller, A.R., Paris, R.B.: On a result related to transformations and summations of generalized hypergeometric series. Math. Commun. 17(1), 205–210 (2012)MathSciNetMATHGoogle Scholar
  18. 18.
    Miller, A.R., Paris, R.B.: Transformation formulas for the generalized hypergeometric function with integral parameter differences. Rocky Mt. J. Math. 43(1), 291–327 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST handbook of mathematical functions. Cambridge University Press, New York (2010)Google Scholar
  20. 20.
    Paszkowski, S.: Convergence acceleration of orthogonal series. Numer. Algorithms 47(1), 35–62 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes 3rd Edition: the art of scientific computing. Cambridge University Press, New York (2007)MATHGoogle Scholar
  22. 22.
    Rathie, A., Paris, R.: Extension of some classical summation theorems for the generalized hypergeometric series with integral parameter differences. J. Class. Anal. 3, 109–127 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sidi, A.: A new method for deriving padé approximants for some hypergeometric functions. J. Comput. Appl. Math. 7, 37–40 (1981)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Sidi, A.: Practical extrapolation methods—theory and applications, Cambridge monographs on applied and computational mathematics, vol. 10. Cambridge University Press (2003)Google Scholar
  25. 25.
    Slater, L.J.: Generalized hypergeometric functions. Cambridge University Press, Cambridge (1966)MATHGoogle Scholar
  26. 26.
    Smith, D.A., Ford, W.F.: Acceleration of linear and logarithmic convergence. SIAM J. Numer. Anal. 16, 223–240 (1979)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wang, M.K., Chu, Y.M., Song, Y.Q.: Asymptotical formulas for gaussian and generalized hypergeometric functions. Appl. Math. Comput. 276(C), 44–60 (2016)MathSciNetGoogle Scholar
  28. 28.
    Weniger, E.J.: Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep. 10, 189–371 (1989)CrossRefGoogle Scholar
  29. 29.
    Weniger, E.J.: Interpolation between sequence transformations. Numer. Algorithms 3(1–4), 477–486 (1992)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Willis, J.: Acceleration of generalized hypergeometric functions through precise remainder asymptotics. Numer. Algorithms 59(3), 447–485 (2012)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wimp, J.: Sequence transformations and their applications. In: Mathematics in Science and Engineering, vol. 154. Academic Press, New York (1981)Google Scholar
  32. 32.
    Woźny, P.: Efficient algorithm for summation of some slowly convergent series. Appl. Numer. Math. 60(12), 1442–1453 (2010)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Woźny, P., Nowak, R.: Method of summation of some slowly convergent series. Appl. Math. Comput. 215(4), 1622–1645 (2009)MathSciNetMATHGoogle Scholar
  34. 34.
    Wynn, P.: On a device for computing the e m(S n) transformation. Math. Tables Aids Comput. 10, 91–96 (1956)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Zucker, I.J., Joyce, G.S.: Special values of the hypergeometric series II. Math. Proc. Cambridge 131, 309–319 (2001)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland

Personalised recommendations