Abstract
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.
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References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 10th Printing Edn. National Bureau of Standards, Washington (1972)
Aitken, A.C.: On Bernoulli’s numerical solution of algebraic equations. Proc. Roy. Soc. Edinburgh 46, 289–305 (1926)
Andrews, G.E., Askey, R., Roy, R.: Special functions. Cambridge University Press, Cambridge (1999)
Baker, Jr, G.A.: Essentials of Padé approximants. Academic Press, New York (1975)
Baker Jr, G.A., Graves-Morris, P.: Padé approximants. Part I: basic theory. II: Extensions and applications. Addison-Wesley, Reading (1981)
Brezinski, C.: Convergence acceleration during the 20th century. J. Comput. Appl. Math. 122(1–2), 1–21 (2000)
Brezinski, C., Redivo Zaglia, M.: Extrapolation methods: theory and practice, studies in computational mathematics, vol. 2, North-Holland (1991)
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)
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)
Homeier, H.H.H.: A hierarchically consistent, iterative sequence transformation. Numer. Algorithms 8(1), 47–81 (1994)
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)
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)
Levin, D.: Development of non-linear transformations for improving convergence of sequences. J. Comput. Math. 3, 371–388 (1973)
Lewanowicz, S., Paszkowski, S.: An analytic method for convergence acceleration of certain hypergeometric series. Math. Comput. 64(210), 691–713 (1995)
Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and theorems for the special functions of mathematical physics. Springer, New York (1966)
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)
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)
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)
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)
Paszkowski, S.: Convergence acceleration of orthogonal series. Numer. Algorithms 47(1), 35–62 (2008)
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)
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)
Sidi, A.: A new method for deriving padé approximants for some hypergeometric functions. J. Comput. Appl. Math. 7, 37–40 (1981)
Sidi, A.: Practical extrapolation methods—theory and applications, Cambridge monographs on applied and computational mathematics, vol. 10. Cambridge University Press (2003)
Slater, L.J.: Generalized hypergeometric functions. Cambridge University Press, Cambridge (1966)
Smith, D.A., Ford, W.F.: Acceleration of linear and logarithmic convergence. SIAM J. Numer. Anal. 16, 223–240 (1979)
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)
Weniger, E.J.: Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep. 10, 189–371 (1989)
Weniger, E.J.: Interpolation between sequence transformations. Numer. Algorithms 3(1–4), 477–486 (1992)
Willis, J.: Acceleration of generalized hypergeometric functions through precise remainder asymptotics. Numer. Algorithms 59(3), 447–485 (2012)
Wimp, J.: Sequence transformations and their applications. In: Mathematics in Science and Engineering, vol. 154. Academic Press, New York (1981)
Woźny, P.: Efficient algorithm for summation of some slowly convergent series. Appl. Numer. Math. 60(12), 1442–1453 (2010)
Woźny, P., Nowak, R.: Method of summation of some slowly convergent series. Appl. Math. Comput. 215(4), 1622–1645 (2009)
Wynn, P.: On a device for computing the e m(S n) transformation. Math. Tables Aids Comput. 10, 91–96 (1956)
Zucker, I.J., Joyce, G.S.: Special values of the hypergeometric series II. Math. Proc. Cambridge 131, 309–319 (2001)
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Nowak, R., Woźny, P. New properties of a certain method of summation of generalized hypergeometric series. Numer Algor 76, 377–391 (2017). https://doi.org/10.1007/s11075-016-0261-1
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DOI: https://doi.org/10.1007/s11075-016-0261-1