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 TMF, 2006, Volume 146, Number 3, Pages 355–364 (Mi tmf2040)

Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation

D. P. Novikov

Omsk State Technical University

Abstract: We show that under the Euler integral transformation with the kernel $(x-z)^{-\alpha}$, some solutions of the Fuchs equations (the original pair for the Painlevé VI equation) pass into solutions of a system of the same form with the parameters changed according to the Okamoto transformation.

Keywords: Painlevé VI equation, Heun equation, Euler integral transformation, Schlesinger transformation, Okamoto transformation

DOI: https://doi.org/10.4213/tmf2040

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English version:
Theoretical and Mathematical Physics, 2006, 146:3, 295–303

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Revised: 18.08.2005

Citation: D. P. Novikov, “Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation”, TMF, 146:3 (2006), 355–364; Theoret. and Math. Phys., 146:3 (2006), 295–303

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. Yu. Slavyanov, F. R. Vukailovich, “Isomonodromic deformations and “antiquantization” for the simplest ordinary differential equations”, Theoret. and Math. Phys., 150:1 (2007), 123–131
2. Joshi N, Kitaev AV, Treharne PA, “On the linearization of the Painlevé III-VI equations and reductions of the three-wave resonant system”, Journal of Mathematical Physics, 48:10 (2007), 103512
3. Haraoka Y, Filipuk G, “Middle convolution and deformation for Fuchsian systems”, Journal of the London Mathematical Society-Second Series, 76:Part 2 (2007), 438–450
4. A. Ya. Kazakov, S. Yu. Slavyanov, “Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation”, Theoret. and Math. Phys., 155:2 (2008), 722–733
5. Kouichi Takemura, “Middle Convolution and Heun's Equation”, SIGMA, 5 (2009), 040, 22 pp.
6. Filipuk, GV, “A hypergeometric system of the Heun equation and middle convolution”, Journal of Physics A-Mathematical and Theoretical, 42:17 (2009), 175208
7. 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
8. 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
9. Nagoya H., “Fractional Calculus of Quantum Painlevé Systems of Type _ ???”, Algebraic and Analytic Aspects of Integrable Systems and Painlev? Equations, Contemporary Mathematics, 651, ed. Dzhamay A. Maruno K. Ormerod C., Amer Mathematical Soc, 2015, 39–64
10. Takemura K., “Integral Transformation of Heun'S Equation and Some Applications”, J. Math. Soc. Jpn., 69:2 (2017), 849–891
11. Farrokh Atai, Edwin Langmann, “Series Solutions of the Non-Stationary Heun Equation”, SIGMA, 14 (2018), 011, 32 pp.
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