Mathematical Notes

, Volume 103, Issue 3–4, pp 357–371 | Cite as

Representations of the Klein Group Determined by Quadruples of Polynomials Associated with the Double Confluent Heun Equation

  • V. M. Buchstaber
  • S. I. Tertychnyi


The canonical representation of the Klein group K4 = ℤ2⊕ℤ2 on the space ℂ* = ℂ {0} induces a representation of this group on the ring L = C[z, z−1], z ∈ ℂ*, of Laurent polynomials and, as a consequence, a representation of the group K4 on the automorphism group of the group G = GL(4,L) by means of the elementwise action. The semidirect product ĜG = GK4 is considered together with a realization of the group Ĝ as a group of semilinear automorphisms of the free 4-dimensional L-module M4. A three-parameter family of representations R of K4 in the group Ĝ and a three-parameter family of elements X ∈ M4 with polynomial coordinates of degrees 2( − 1), 2, 2( − 1), and 2, where is an arbitrary positive integer (one of the three parameters), are constructed. It is shown that, for any given family of parameters, the vector X is a fixed point of the corresponding representation R. An algorithm for calculating the polynomials that are the components of X was obtained in a previous paper of the authors, in which it was proved that these polynomials give explicit formulas for automorphisms of the solution space of the doubly confluent Heun equation.


semilinear mappings ring of Laurent polynomials representations of the Klein group double confluent Heun equation 


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  1. 1.
    V. M. Bukhshtaber and S. I. Tertychnyi, “Automorphisms of the solution spaces of special double-confluent Heun equations,” Funktsional. Anal. i Prilozhen. 50 (3), 12–33 (2016) [Functional Anal. Appl. 50 (3), 176–192 (2016)].MathSciNetCrossRefGoogle Scholar
  2. 2.
    D. Schmidt and G. Wolf, “The double confluent Heun equation,” in Heun’s Differential Equations, Ed. by A. Ronveaux (Oxford Univ. Press, New York, 1995).Google Scholar
  3. 3.
    S. Slavyanov and W. Lay, Special Functions. Unified Theory Based on Singularities (Oxford University Press, Oxford, 2000; Nevskii Dialekt, St. Petersburg., 2002).zbMATHGoogle Scholar
  4. 4.
    V. M. Bukhshtaber [Buchstaber] and S. I. Tertychnyi, “Holomorphis solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction,” Teoret.Mat. Fiz. 182 (3), 373–404 (2015) [Theoret. and Math. Phys. 182 (3), 329–355 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Klein Four-Group, Scholar
  6. 6.
    J. Dieudonné, La géométrie des groupes classiques (Springer-Verlag, Berlin–New York, 1971; Mir, Moscow, 1974).zbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.All-Russian Scientific Research Institute of Physical, Technical, and Radiotechnical MeasurementsMendeleevo, Moscow OblastRussia

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