Programming and Computer Software

, Volume 34, Issue 4, pp 187–190 | Cite as

On the bottom summation

Article
  • 20 Downloads

Abstract

We consider summation of consecutive values (φ(v), φ(v + 1), ..., φ(w) of a meromorphic function φ(z), where v, w ∈ ℤ. We assume that φ(z) satisfies a linear difference equation L(y) = 0 with polynomial coefficients, and that a summing operator for L exists (such an operator can be found—if it exists—by the Accurate Summation algorithm, or, alternatively, by Gosper’s algorithm when ordL = 1). The notion of bottom summation which covers the case where φ(z) has poles in ℤ is introduced.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramov, S.A. and Petkovšek, M., Analytic Solutions of Linear Difference Equations, Formal Series, and Bottom Summation, Proc. Computer Algebra in Scientific Computing, 10th Int. Workshop, CASC’07 (Bonn, Germany, 2007), Lecture Notes in Computer Science, 2007, vol. 4770, pp. 1–10.CrossRefGoogle Scholar
  2. 2.
    Abramov, S.A. and van Hoeij, M., Set of Poles of Solutions of Linear Difference Equations with Polynomial Coefficients, Zh. Vychisl. Mat. Mat. Fiz., 2003, vol. 43, no. 1, pp. 60–65 [J. Comput. Math. Math. Phys., 2003, vol. 43, no. 1, pp. 57–62].MathSciNetGoogle Scholar
  3. 3.
    Gosper, R.W., Jr., Decision Procedure for Indefinite Hypergeometric Summation, Proc. Natl. Acad. Sci. USA, 1978, vol. 75, pp. 40–42.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Abramov, S.A. and van Hoeij, M., Integration of Solutions of Linear Functional Equations, Integral Transforms Special Functions, 1999, vol. 8, pp. 3–12.MATHCrossRefGoogle Scholar
  5. 5.
    Barkatou, M.A., Contribution à l’étude des équations différentielles et aux différences dans le champ complexe, PhD Dissertation, Grenoble, France: INPG, 1989.Google Scholar

Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  1. 1.Dorodnicyn Computing CentreRussian Academy of SciencesMoscowRussia
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

Personalised recommendations