Mathematics in Computer Science

, Volume 4, Issue 2–3, pp 259–266 | Cite as

A Fast Approach to Creative Telescoping

  • Christoph Koutschan


In this note we reinvestigate the task of computing creative telescoping relations in differential–difference operator algebras. Our approach is based on an ansatz that explicitly includes the denominators of the delta parts. We contribute several ideas of how to make an implementation of this approach reasonably fast and provide such an implementation. A selection of examples shows that it can be superior to existing methods by a large factor.


Holonomic functions Special functions Symbolic integration Symbolic summation Creative telescoping Ore algebra WZ theory 

Mathematics Subject Classification (2010)

Primary 68W30 Secondary 33F10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramov, S.A., Barkatou, M.: Rational solutions of first order linear difference systems. In: ISSAC’98: Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation, pp. 124–131. ACM, New York (1998)Google Scholar
  2. 2.
    Andrews G., Askey R., Roy R.: Special functions, Encyclopedia of mathematics and its applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  3. 3.
    Andrews G.E., Paule P.: Some questions concerning computer-generated proofs of a binomial double-sum identity. J. Symb. Comput. 16, 147–153 (1993)zbMATHCrossRefGoogle Scholar
  4. 4.
    Barkatou M.: On rational solutions of systems of linear differential equations. J. Symb. Comput. 28, 547–567 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chyzak F.: An extension of Zeilberger’s fast algorithm to general holonomic functions. Discret. Math. 217(1–3), 115–134 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chyzak, F., Kauers, M., Salvy, B.: A non-holonomic systems approach to special function identities. In: ISSAC’09: Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, pp. 111–118. ACM, New York (2009)Google Scholar
  7. 7.
    Chyzak F., Salvy B.: Non-commutative elimination in ore algebras proves multivariate identities. J. Symb. Comput. 26, 187–227 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kauers, M., Koutschan, C., Zeilberger, D.: A proof of George Andrews’ and Dave Robbins’ q-TSPP conjecture (modulo a finite amount of routine calculations). The personal journal of Shalosh B. Ekhad and Doron Zeilberger, pp. 1–8. (2009)
  9. 9.
    Klein, S.: Heavy flavor coefficient functions in deep-inelastic scattering at O(a s2) and large virtualities. Diplomarbeit, Universität Potsdam, Germany (2006)Google Scholar
  10. 10.
    Koutschan, C.: Advanced applications of the holonomic systems approach. PhD Thesis, RISC, Johannes Kepler University, Linz, Austria (2009)Google Scholar
  11. 11.
    Koutschan, C.: Eliminating human insight: an algorithmic proof of Stembridge’s TSPP theorem. In: Amdeberhan, T., Medina, L., Moll, V. (eds.) Gems in Experimental Mathematics, Contemporary Mathematics, vol. 517, pp. 219–230. American Mathematical Society (2010)Google Scholar
  12. 12.
    Koutschan, C., Kauers, M., Zeilberger, D.: A proof of George Andrews’ and David Robbins’ q-TSPP conjecture. Technical Report (2010). arXiv:1002.4384Google Scholar
  13. 13.
    Takayama, N.: An algorithm of constructing the integral of a module—an infinite dimensional analog of Gröbner basis. In: ISSAC’90: Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. 206–211. ACM, New York (1990)Google Scholar
  14. 14.
    Takayama, N.: Gröbner basis, integration and transcendental functions. In: ISSAC’90: Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. 152–156. ACM, New York (1990)Google Scholar
  15. 15.
    Wegschaider, K.: Computer generated proofs of binomial multi-sum identities. Master’s Thesis, RISC, Johannes Kepler University Linz (1997)Google Scholar
  16. 16.
    Wilf H.S., Zeilberger D.: An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent. Math. 108(1), 575–633 (1992)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Zeilberger D.: A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32(3), 321–368 (1990)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of MathematicsTulane UniversityNew OrleansUSA

Personalised recommendations