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On a summability method

Application Of Padé Approximation

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Part of the Lecture Notes in Mathematics book series (LNM,volume 765)

Abstract

In previous studies we applied Lanzcos' τ-method to get polynomial and rational approximations to series of hypergeometric type. It was shown that the approximations could be viewed as a weighted sum of the partial sums of the given series. This we call a summability method. The τ-method starts from the differential equation satisfied by the given function. But in view of the summability feature, the approximations can be found directly from the hypergeometric series. In the present paper, we explore the applicability of the direct summability process for functions of hypergeometric type to functions which are not of hypergeometric type. Our treatment is expository. We treat two examples. The first converges in |z|<2. The second is the divergent but asymptotic series related to the logarithm of the gamma function. Main diagonal Padé approximations are also constructed. The numerics indicate that all of the approximations converge at least for all z>0. These approximations are quite remarkable in that they appear to converge in a region where the given series upon which they are based are divergent.

Keywords

  • Rational Approximation
  • Polynomial Approximation
  • Hypergeometric Series
  • Summability Method
  • Summability Process

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This research was sponsored by the National Science Foundation under Grant MCS-78-01351.

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References

  1. Y.L. Luke, "The Special Functions and Their Approximations," Vols. 1, 2, Academic Press, Inc., New York, 1969.

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  2. Y.L. Luke, "Mathematical Functions and Their Approximations," Academic Press, Inc., New York, 1975.

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  3. Y.L. Luke, "Algorithms for the Computation of Mathematical Functions," Academic Press, Inc., New York, 1977.

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© 1979 Springer-Verlag

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Luke, Y.L. (1979). On a summability method. In: Wuytack, L. (eds) Padé Approximation and its Applications. Lecture Notes in Mathematics, vol 765. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085587

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  • DOI: https://doi.org/10.1007/BFb0085587

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09717-4

  • Online ISBN: 978-3-540-38511-0

  • eBook Packages: Springer Book Archive