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TMF, 2008, Volume 155, Number 2, Pages 252–264 (Mi tmf6209)  

This article is cited in 19 scientific papers (total in 19 papers)

Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

A. Ya. Kazakova, S. Yu. Slavyanovb

a Saint-Petersburg State University of Aerospace Instrumentation
b Saint-Petersburg State University

Abstract: Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation.

Keywords: Euler transformation, Heun equation, Painlevé equation

DOI: https://doi.org/10.4213/tmf6209

Full text: PDF file (462 kB)
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English version:
Theoretical and Mathematical Physics, 2008, 155:2, 722–733

Bibliographic databases:

Received: 29.10.2007

Citation: A. Ya. Kazakov, S. Yu. Slavyanov, “Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation”, TMF, 155:2 (2008), 252–264; Theoret. and Math. Phys., 155:2 (2008), 722–733

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Kouichi Takemura, “Middle Convolution and Heun's Equation”, SIGMA, 5 (2009), 040, 22 pp.  mathnet  crossref  mathscinet  zmath
    2. Filipuk G.V., “A hypergeometric system of the Heun equation and middle convolution”, J. Phys. A, 42:17 (2009), 175208, 11 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. D. P. Novikov, “The $2{\times}2$ matrix Schlesinger system and the Belavin–Polyakov–Zamolodchikov system”, Theoret. and Math. Phys., 161:2 (2009), 1485–1496  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Kazakov A.Ya., Slavyanov S.Yu., “Integral Symmetries for Confluent Heun Equations and Symmetries of Painlevé Equation P-5”, Painleve Equations and Related Topics (2012), Degruyter Proceedings in Mathematics, eds. Bruno A., Batkhin A., Walter de Gruyter & Co, 2012, 237–239  mathscinet  isi
    5. Slavyanov S.Y., “Relations Between Linear Equations and Painlevé'S Equations”, Constr. Approx., 39:1, SI (2014), 75–83  crossref  mathscinet  zmath  isi  scopus  scopus
    6. A. Ya. Kazakov, S. Yu. Slavyanov, “Euler integral symmetries for the confluent Heun equation and symmetries of the Painlevé equation PV”, Theoret. and Math. Phys., 179:2 (2014), 543–549  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. A. Ya. Kazakov, “Integral symmetry for the confluent Heun equation with added apparent singularity”, J. Math. Sci. (N. Y.), 214:3 (2016), 268–276  mathnet  crossref  mathscinet
    8. S. Yu. Slavyanov, “Polynomial degree reduction of a Fuchsian $2{\times}2$ system”, Theoret. and Math. Phys., 182:2 (2015), 182–188  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. Leroy C., Ishkhanyan A.M., “Expansions of the Solutions of the Confluent Heun Equation in Terms of the Incomplete Beta and the Appell Generalized Hypergeometric Functions”, Integral Transform. Spec. Funct., 26:6 (2015), 451–459  crossref  mathscinet  zmath  isi  scopus  scopus
    10. J. Math. Sci. (N. Y.), 209:6 (2015), 910–921  mathnet  crossref
    11. S. Yu. Slavyanov, “Antiquantization and the corresponding symmetries”, Theoret. and Math. Phys., 185:1 (2015), 1522–1526  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. Shahverdyan T.A., Ishkhanyan T.A., Grigoryan A.E., Ishkhanyan A.M., “Analytic Solutions of the Quantum Two-State Problem in Terms of the Double, Bi- and Triconfluent Heun Functions”, J. Contemp. Phys.-Armen. Acad. Sci., 50:3 (2015), 211–226  crossref  mathscinet  isi  scopus
    13. S. Yu. Slavyanov, D. F. Shat'ko, A. M. Ishkhanyan, T. A. Rotinyan, “Generation and removal of apparent singularities in linear ordinary differential equations with polynomial coefficients”, Theoret. and Math. Phys., 189:3 (2016), 1726–1733  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    14. S. Yu. Slavyanov, O. L. Stesik, “Symbolic generation of Painlevé equations”, J. Math. Sci. (N. Y.), 224:2 (2017), 345–348  mathnet  crossref  mathscinet
    15. Ishkhanyan A.M., “A singular Lambert- W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions”, Mod. Phys. Lett. A, 31:33 (2016), 1650177  crossref  mathscinet  zmath  isi  elib  scopus
    16. Chen Zh., Kuo T.-J., Lin Ch.-Sh., “Hamiltonian system for the elliptic form of Painlevé VI equation”, J. Math. Pures Appl., 106:3 (2016), 546–581  crossref  mathscinet  zmath  isi  elib  scopus
    17. S. Yu. Slavyanov, “Symmetries and apparent singularities for the simplest Fuchsian equations”, Theoret. and Math. Phys., 193:3 (2017), 1754–1760  mathnet  crossref  crossref  adsnasa  isi  elib
    18. Takemura K., “Integral Transformation of Heun'S Equation and Some Applications”, J. Math. Soc. Jpn., 69:2 (2017), 849–891  crossref  mathscinet  zmath  isi  scopus  scopus
    19. Babich M., Slavyanov S., “Antiquantization, Isomonodromy, and Integrability”, J. Math. Phys., 59:9, SI (2018), 091416  crossref  mathscinet  zmath  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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