Skip to main content
SpringerLink
Log in

Symbolic Conversion of Holonomic Functions to Hypergeometric Type Power Series

  • Published:
Programming and Computer Software Aims and scope Submit manuscript

Abstract

A term an is m-fold hypergeometric, for a given positive integer m, if the ratio \({{a}_{{n + m}}}{\text{/}}{{a}_{n}}\) is a rational function over a field \(\mathbb{K}\) of characteristic zero. We establish the structure of holonomic recurrence equations, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have m-fold hypergeometric term solutions over \(\mathbb{K}\), for any positive integer m. Consequently, we describe a new algorithm, say mfoldHyper, that extends the algorithms by Petkovšek (1992) and van Hoeij (1998) which compute a basis of hypergeometric (m = 1) term solutions of holonomic recurrence equations to the more general case of m-fold hypergeometric terms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Mostly \(\mathbb{K}: = \mathbb{Q}({{\alpha }_{1}}, \ldots ,{{\alpha }_{N}})\) is the field of rational functions in several variables

  2. Originally called hypergeometric type but we avoid this calling since we are redefining this terminology.

  3. Throughout this paper we mainly give representations about z0 = 0 since the case of arbitrary z0 deduces easily.

  4. A Maple command to use package names as Maple procedures.

  5. The tangent function is usually not encoded as \(\sin (z){\text{/cos}}(z)\), and this fact is also ignore by the implemented differentiation.

  6. For each fixed m corresponding to an m-fold hypergeometric term we have a basis, and the basis of all m-fold hypergeometric term solutions is the collection of these bases.

  7. The brackets around m,j means optional arguments

  8. Originally the type was used to denote the value of \(m\) for an \(m\)‑fold hypergeometric term coefficient.

  9. Integer shift used in Petkovšek’s algorithm, see also Lemma 8

  10. This is to make sure that cancellation of common factors is avoided.

REFERENCES

  1. Abramov, S.A., m-sparse solutions of linear ordinary differential equations with polynomial coefficients, Discrete Math., 2000, vol. 217, no. 1–3, pp. 3–15.

  2. Abramov, S.A., Petkovšek, M., and Ryabenko, A., Special formal series solutions of linear operator equations, Discrete Math., 2000, vol. 210, no. 1–3, pp. 3–25.

  3. Almkvist, G. and Zeilberger, D., The method of differentiating under the integral sign, J. Symb. Comput., 1990, vol. 10, no. 6, pp. 571–592.

    Article  MathSciNet  Google Scholar 

  4. Brewer, T., Algebraic properties of formal power series composition, Ph.D. Thesis, Univ. of Kentucky, 2014. https://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1021&context=math_etds.

  5. Cluzeau, T. and van Hoeij, M., Computing hypergeometric solutions of linear recurrence equations, Appl. Algebra Eng. Comm. Comput., 2006, vol. 17, no. 2, pp. 83–115.

    Article  MathSciNet  Google Scholar 

  6. Gruntz, D. and Koepf, W., Maple package on formal power series, Maple Tech. Newsl., 1995, vol. 2, no. 2, pp. 22–28.

    Google Scholar 

  7. Hendricks, P.A. and Singer, M.F., Solving difference equations in finite terms, J. Symb. Comput., 1999, vol. 27, vol. 3, pp. 239–259.

  8. Horn, P., Koepf, W., and Sprenger, T., m-fold hypergeometric solutions of linear recurrence equations revisited, Math. Comput. Sci., 2012, vol. 6, no. 1, pp. 61–77.

    Article  MathSciNet  Google Scholar 

  9. Kauers, M. and Paule, P., The Concrete Tetrahedron. Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates, Wien: Springer-Verlag, 2011.

    MATH  Google Scholar 

  10. Koepf, W., Power series in computer algebra, J. Symb. Comput., 1992, vol. 13, no. 6, pp. 581–603.

    Article  MathSciNet  Google Scholar 

  11. Koepf, W., The algebra of holonomic equations, Math. Semesterber, 1997, vol. 44, pp. 173–194.

    Article  MathSciNet  Google Scholar 

  12. Koepf, W. and Schmersau, D., Representations of orthogonal polynomials, J. Comput. Appl. Math., 1998, vol. 90, no. 1, pp. 57–94.

    Article  MathSciNet  Google Scholar 

  13. Lubin, J., Nonarchimedean dynamical systems, Compos. Math., 1994, vol. 94, no. 3, pp. 321–346.

    MATH  Google Scholar 

  14. Maplesoft, a. d. o. W. M. I., 2020. Maple 2021.

  15. Petkovšek, M., Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comput., 1992, vol. 14, no. 2-3, pp. 243–264.

    Article  MathSciNet  Google Scholar 

  16. Petkovšek, M. and Salvy, B., Finding all hypergeometric solutions of linear differential equations, in ISSAC, Bronstein, M., Ed., New York: Association for Computing Machinery, 1993, pp. 27–33.

    Google Scholar 

  17. Ryabenko, A., Special formal series solutions of linear ordinary differential equations, in Formal Power Series and Algebraic Combinatorics, Springer, 2000, pp. 356–366.

    MATH  Google Scholar 

  18. Ryabenko, A.A., Formal solutions of linear ordinary differential equations containing m-hypergeometric series, Program. Comput. Software, 2002, vol. 28, no. 2, pp. 92–101.

    Article  MathSciNet  Google Scholar 

  19. Salvy, B. and Zimmermann, P., Gfun: a maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software, 1994, vol. 20, no. 2, pp. 163–177.

    Article  Google Scholar 

  20. Stanley, R.P., Enumerative Combinatorics, 2nd ed., Cambridge, 2011, vol. 1.

    Book  Google Scholar 

  21. Teguia Tabuguia, B., Power series representations of hypergeometric type and non-holonomic functions in computer algebra, Ph.D. Thesis, Univ. of Kassel, 2020. https://kobra.uni-kassel.de/handle/123456789/11598.

  22. Teguia Tabuguia, B., A variant of van Hoeij’s algorithm to compute hypergeometric term solutions of holonomic recurrence equations, J. Algorithm Comput., 2021, vol. 53, no. 2, pp. 1–32.

  23. Teguia Tabuguia, B. and Koepf, W., Hypergeometric type power series, in Proc. 4th Int. Conf. “Computer Algebra”, Abramov, S.A. and Sevastyanov, L.A., Eds., MAKS Press, 2021, vol. 28v pp. 105–109.

  24. Teguia Tabuguia, B. and Koepf, W., Power series representations of hypergeometric type functions, in Maple in Mathematics Education and Research. MC 2020, Corless, R., Gerhard, J., and Kotsireas, I., Eds., Springer, 2021, pp. 376–393.

    Google Scholar 

  25. Van Hoeij, M., Finite singularities and hypergeometric solutions of linear recurrence equations, J. Pure Appl. Algebra, 1999, vol. 139, no. 1–3, pp. 109–131.

  26. Wolfram, S., The Mathematica Book, Version 4, 5th ed., Wolfram Media, Cambridge University Press, 2003.

  27. Zariski, O. and Samuel, P., Commutative Algebra, New York: Springer-Verlag, 1960, Vol. 2.

    Book  Google Scholar 

  28. Zeilberger, D., A holonomic systems approach to special functions identities, J. Comput. Appl. Math., 1990, vol. 32, no. 3, pp. 321–368.

    Article  MathSciNet  Google Scholar 

  29. Teguia Tabuguia, B. and Koepf, W., On the representation of non-holonomic power series, 2021. arXiv:2109.09574 [cs.SC].

Download references

ACKNOWLEDGMENTS

We thank Anna Ryabenko and Sergei Abramov who gave the opportunity to take part in the 4th international conference “Computer Algebra”, Moscow, June 28-29, 2021, and handled our submission to the Russian journal of Programming and Computer Software.

We also thank Jürgen or Gerhard from Maplesoft for the opportunity to implement our algorithms for the Maple 2022 release.

We would like to thank the reviewers for their important help. This gave us some insights that could be incorporated into the final version.

Author information

Authors and Affiliations

  1. University of Kassel, 34132, Kassel, Germany

    Bertrand Teguia Tabuguia & Wolfram Koepf

Authors
  1. Bertrand Teguia Tabuguia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wolfram Koepf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Bertrand Teguia Tabuguia or Wolfram Koepf.

Ethics declarations

The authors declare that they have no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Teguia Tabuguia, B., Koepf, W. Symbolic Conversion of Holonomic Functions to Hypergeometric Type Power Series. Program Comput Soft 48, 125–146 (2022). https://doi.org/10.1134/S0361768822020104

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768822020104

Search

Navigation