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Computational problems of multivariate hypergeometric theory. (English. Russian original) Zbl 1458.68286
Program. Comput. Softw. 44, No. 2, 131-137 (2018); translation from Programmirovanie 44, No. 2, 74-82 (2018).
Summary: 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.

68W30 Symbolic computation and algebraic computation
33F05 Numerical approximation and evaluation of special functions
Full Text: DOI
[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. · Zbl 1344.34023
[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. · Zbl 1247.65156
[3] Gelfond, A.O., Calculus of Finite Differences, Hindustan Publ. Corp., 1971.
[4] Stanley, R.P., Enumerative Combinatorics, Cambridge University Press, 2010. · Zbl 0889.05001
[5] Sadykov, T. M., On a multidimensional system of hypergeometric differential equations, 986-997 (1998)
[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. · Zbl 1159.33003
[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.
[8] Krasikov, V.A. and Sadykov, T.M., On the analytic complexity of discriminants, Proc. Steklov Inst. Math., 2012, vol. 279, pp. 78-92. · Zbl 1298.32007
[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.
[10] Sadykov, T.M. and Tanabé, S., Maximally reducible monodromy of bivariate hypergeometric systems, Izv.: Math., 2016, vol. 80, no. 1, pp. 221-262.
[11] Sadykov, T.M. and Tsikh, A.K., Hypergeometric and Algebraic Functions in Several Variables, Moscow: Nauka, 2014 (in Russian).
[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.
[13] Bousquet-Mélou, M. and Petkovšek, M., Linear recurrences with constant coefficients: the multivariate case, Discrete Math., 2000, vol. 225, pp. 51-75.
[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.
[15] Cattani, E., Dickenstein, A., and Sturmfels, B., Rational hypergeometric functions, Compositio Mathematica, 2001, vol. 128, no. 2, pp. 217-240.
[16] Grayson, D.R. and Stillman, M.E., Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/.
[17] Noro, M., A computer algebra system: Risa/Asir, 147-162 (2003)
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