Programming and Computer Software

, Volume 44, Issue 2, pp 131–137 | Cite as

Computational Problems of Multivariate Hypergeometric Theory



We consider computational problems of the theory of hypergeometric functions in several complex variables: computation of the holonomic rank of a hypergeometric system of partial differential equations, computing the defining polynomial of the singular hypersurface of such a system and finding its monomial solutions. The presented algorithms have been implemented in the computer algebra system MATHEMATICA.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramov, S.A., Search of rational solutions to differential and difference systems by means of formal series, Program. Comput. Software, 2015, vol. 42, no. 2, pp. 65–73.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Abramov, S.A., Gheffar, A., and Khmelnov, D.E., Rational solutions of linear difference equations: Universal denominators and denominator bounds, Program. Comput. Software, 2011, vol. 37, no. 2, pp. 78–86.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Gelfond, A.O., Calculus of Finite Differences, Hindustan Publ. Corp., 1971.Google Scholar
  4. 4.
    Stanley, R.P., Enumerative Combinatorics, Cambridge University Press, 2010.MATHGoogle Scholar
  5. 5.
    Sadykov, T.M., On a multidimensional system of hypergeometric differential equations, Sib. Math. J., 1998, pp. 986–997.Google Scholar
  6. 6.
    Dickenstein, A. and Sadykov, T.M., Bases in the solution space of the Mellin system, Sb.: Math., 2007, vol. 198, no. 9, pp. 1277–1298.MathSciNetMATHGoogle Scholar
  7. 7.
    Dickenstein, A. and Sadykov, T.M., Algebraicity of solutions to the Mellin system and its monodromy, Dokl. Math., 2007, vol. 75, no. 1, pp. 80–82.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Krasikov, V.A. and Sadykov, T.M., On the analytic complexity of discriminants, Proc. Steklov Inst. Math., 2012, vol. 279, pp. 78–92.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kulikov, V.R. and Stepanenko, V.A., On solutions and Waring’s formulas for the system of algebraic equations with unknowns, St. Petersburg Math. J., 2015, vol. 26, no. 5, pp. 839–848.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Sadykov, T.M. and Tanabé, S., Maximally reducible monodromy of bivariate hypergeometric systems, Izv.: Math., 2016, vol. 80, no. 1, pp. 221–262.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Sadykov, T.M. and Tsikh, A.K., Hypergeometric and Algebraic Functions in Several Variables, Moscow: Nauka, 2014 (in Russian).MATHGoogle Scholar
  12. 12.
    Abramov, S.A., Barkatou, M.A., van Hoeij, M., and Petkovsek, M., Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences, J. Symbolic Comput., 2011, vol. 46, no. 11, pp. 1205–1228.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bousquet-Mélou, M. and Petkovšek, M., Linear recurrences with constant coefficients: the multivariate case, Discrete Math., 2000, vol. 225, pp. 51–75.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cattani, E., Dickenstein, A., and Rodriguez Villegas, F., The structure of bivariate rational hypergeometric functions, Int. Math. Res. Notices, 2011, no. 11, pp. 2496–2533.MathSciNetMATHGoogle Scholar
  15. 15.
    Cattani, E., Dickenstein, A., and Sturmfels, B., Rational hypergeometric functions, Compositio Mathematica, 2001, vol. 128, no. 2, pp. 217–240.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Grayson, D.R. and Stillman, M.E., Macaulay2, a software system for research in algebraic geometry. Scholar
  17. 17.
    Noro, M., A computer algebra system: Risa/Asir, Algebra, Geometry and Software Systems, 2003, pp. 147–162.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Plekhanov Russian UniversityMoscowRussia
  2. 2.Siberian Federal UniversityKrasnoyarskRussia

Personalised recommendations