Advertisement

On the analytic complexity of hypergeometric functions

  • T. M. Sadykov
Article
  • 24 Downloads

Abstract

Hypergeometric functions of several variables resemble functions of finite analytic complexity in the sense that the elements of both classes satisfy certain canonical overdetermined systems of partial differential equations. Otherwise these two sets of functions are very different. We investigate the relation between the two classes of functions and compute the analytic complexity of certain bivariate hypergeometric functions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. I. Arnold, “Representation of continuous functions of three variables by the superposition of continuous functions of two variables,” Mat. Sb. 48 (1), 3–74 (1959) [Am. Math. Soc. Transl., Ser. 2, 28, 61–147 (1963)].MathSciNetGoogle Scholar
  2. 2.
    V. K. Beloshapka, “Analytic complexity of functions of two variables,” Russ. J. Math. Phys. 14 (3), 243–249 (2007).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    V. K. Beloshapka, “Analytical complexity: Development of the topic,” Russ. J. Math. Phys. 19 (4), 428–439 (2012).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    V. K. Beloshapka, “Analytic complexity of functions of several variables,” Mat. Zametki 100 (6), 781–789 (2016) [Math. Notes 100, 774–780 (2016)].MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    V. K. Beloshapka, “Algebraic functions of complexity one, a Weierstrass theorem, and three arithmetic operations,” Russ. J. Math. Phys. 23 (3), 343–347 (2016).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    V. K. Beloshapka, “Three families of functions of complexity one,” J. Sib. Fed. Univ., Math. Phys. 9 (4), 416–426 (2016).MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Dickenstein, L. F. Matusevich, and E. Miller, “Binomial D-modules,” Duke Math. J. 151 (3), 385–429 (2010).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    A. Dickenstein and T. M. Sadykov, “Algebraicity of solutions to the Mellin system and its monodromy,” Dokl. Akad. Nauk 412 (4), 448–450 (2007) [Dokl. Math. 75 (1), 80–82 (2007)].MATHGoogle Scholar
  9. 9.
    A. Dickenstein and T. M. Sadykov, “Bases in the solution space of the Mellin system,” Mat. Sb. 198 (9), 59–80 (2007) [Sb. Math. 198, 1277–1298 (2007)].MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    I. M. Gel’fand, M. I. Graev, and V. S. Retakh, “General hypergeometric systems of equations and series of hypergeometric type,” Usp. Mat. Nauk 47 (4), 3–82 (1992) [Russ. Math. Surv. 47 (4), 1–88 (1992)].MathSciNetMATHGoogle Scholar
  11. 11.
    M. Yu. Kalmykov and B. A. Kniehl, “Mellin–Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions,” Phys. Lett. B 714 (1), 103–109 (2012).MathSciNetCrossRefGoogle Scholar
  12. 12.
    V. A. Krasikov and T. M. Sadykov, “On the analytic complexity of discriminants,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 279, 86–101 (2012) [Proc. Steklov Inst. Math. 279, 78–92 (2012)].MathSciNetMATHGoogle Scholar
  13. 13.
    V. P. Palamodov, Linear Differential Operators with Constant Coefficients (Springer, Berlin, 1970).CrossRefMATHGoogle Scholar
  14. 14.
    T. M. Sadykov, “On a multidimensional system of hypergeometric differential equations,” Sib. Mat. Zh. 39 (5), 1141–1153 (1998) [Sib. Math. J. 39, 986–997 (1998)].CrossRefMATHGoogle Scholar
  15. 15.
    T. M. Sadykov, “On the Horn system of partial differential equations and series of hypergeometric type,” Math. Scand. 91 (1), 127–149 (2002).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    T. Sadykov, “The Hadamard product of hypergeometric series,” Bull. Sci. Math. 126 (1), 31–43 (2002).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    T. M. Sadykov, “Hypergeometric systems of equations with maximally reducible monodromy,” Dokl. Akad. Nauk 423 (4), 455–457 (2008) [Dokl. Math. 78 (3), 880–882 (2008)].MathSciNetMATHGoogle Scholar
  18. 18.
    T. M. Sadykov and S. Tanabé, “Maximally reducible monodromy of bivariate hypergeometric systems,” Izv. Ross. Akad. Nauk, Ser. Mat. 80 (1), 235–280 (2016) [Izv. Math. 80, 221–262 (2016)].MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    M. Sato, T. Shintani, and M. Muro, “Theory of prehomogeneous vector spaces (algebraic part),” Nagoya Math. J. 120, 1–34 (1990).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Plekhanov Russian University of EconomicsMoscowRussia

Personalised recommendations