Advertisement

A q-Analogue of the Modified Abramov-Petkovšek Reduction

  • Hao DuEmail author
  • Hui Huang
  • Ziming Li
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 226)

Abstract

We present an additive decomposition algorithm for q-hypergeometric terms. It decomposes a given term T as the sum of two terms, in which the former is q-summable and the latter is minimal in some technical sense. Moreover, the latter is zero if and only if T is q-summable. Although our additive decomposition is a q-analogue of the modified Abramov-Petkovšek reduction for usual hypergeometric terms, they differ in some subtle details. For instance, we need to reduce Laurent polynomials instead of polynomials in the q-case. The experimental results illustrate that the additive decomposition is more efficient than q-Gosper’s algorithm for determining q-summability when some q-dispersion concerning the input term becomes large. Moreover, the additive decomposition may serve as a starting point to develop a reduction-based creative-telescoping method for q-hypergeometric terms.

Keywords

Additive decomposition q-hypergeometric term Reduction Symbolic summation 

Notes

Acknowledgements

We thank the anonymous referee for helpful comments and valuable references.

References

  1. 1.
    Abramov, S.A.: The rational component of the solution of a first order linear recurrence relation with a rational right hand side. Ž. Vyčisl. Mat. i Mat. Fiz. 15(4), 1035–1039, 1090 (1975)Google Scholar
  2. 2.
    Abramov, S.A., Petkovšek, M.: Minimal decomposition of indefinite hypergeometric sums. In: ISSAC 2001—Proceedings of the 26th International Symposium on Symbolic and Algebraic Computation, pp. 7–14 (electronic). ACM, New York (2001)Google Scholar
  3. 3.
    Abramov, S.A., Petkovšek, M.: Rational normal forms and minimal decompositions of hypergeometric terms. J. Symbolic Comput. 33(5), 521–543 (2002). Computer algebra, London, ON (2001)Google Scholar
  4. 4.
    Bostan, A., Chen, S., Chyzak, F., Li, Z., Xin, G.: Hermite reduction and creative telescoping for hyperexponential functions. In: ISSAC 2013—Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, pp. 77–84. ACM, New York (2013)Google Scholar
  5. 5.
    Bronstein, M., Li, Z., Wu, M.: Picard-Vessiot extensions for linear functional systems. In: ISSAC 2005—Proceedings of the 30th International Symposium on Symbolic and Algebraic Computation, pp. 68–75. ACM, New York (2005)Google Scholar
  6. 6.
    Chen, S., Huang, H., Kauers, M., Li, Z.: A modified Abramov-Petkovšek reduction and creative telescoping for hypergeometric terms. In: ISSAC 2015—Proceedings of the 40th International Symposium on Symbolic and Algebraic Computation, pp. 117–124. ACM, New York (2015)Google Scholar
  7. 7.
    Chen, S., Kauers, M., Koutschan, C.: Reduction-based creative telescoping for algebraic functions. In: ISSAC 2016—Proceedings of the 41st International Symposium on Symbolic and Algebraic Computation. ACM, New York (2016, to appear)Google Scholar
  8. 8.
    Chen, S., van Hoeij, M., Kauers, M., Koutschan, C.: Reduction-based creative telescoping for Fuchsian D-finite functions. J. Symbolic Comput. (to appear)Google Scholar
  9. 9.
    Chen, W.Y.C., Hou, Q.-H., Mu, Y.-P.: Applicability of the \(q\)-analogue of Zeilberger’s algorithm. J. Symbolic Comput. 39(2), 155–170 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hardouin, C., Singer, M.F.: Differential Galois theory of linear difference equations. Math. Ann. 342(2), 333–377 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Karr, M.: Summation in finite terms. J. Assoc. Comput. Mach. 28(2), 305–350 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Koornwinder, T.H.: On Zeilberger’s algorithm and its q-analogue. J. Comput. Appl. Math. 48(1), 91–111 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Paule, P.: Greatest factorial factorization and symbolic summation. J. Symbolic Comput. 20(3), 235–268 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Paule, P., Riese, A.: A Mathematica \(q\)-analogue of Zeilberger’s algorithm based on an algebraically motivated approach to \(q\)-hypergeometric telescoping. In: Special Functions, \(q\)-Series and Related Topics, Toronto, ON, 1995. Fields Institute Communications, vol. 14, pp. 179–210 (1997). Amer. Math. Soc., Providence, RI (1997)Google Scholar
  15. 15.
    Schneider, C.: Simplifying sums in \({\Pi }{\varSigma }^*\)-extensions. J. Algebra Appl. 6(3), 415–441 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.KLMMAMSS, Chinese Academy of SciencesBeijingChina
  2. 2.Institute for AlgebraJohannes Kepler UniversityLinzAustria
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

Personalised recommendations