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 SIGMA, 2009, Volume 5, 040, 22 pages (Mi sigma386)

Middle Convolution and Heun's Equation

Kouichi Takemura

Department of Mathematical Sciences, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama 236-0027, Japan

Abstract: Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of initial conditions of the sixth Painlevé equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun's equation.

Keywords: Heun's equation; the space of initial conditions; the sixth Painlevé equation; middle convolution

DOI: https://doi.org/10.3842/SIGMA.2009.040

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ArXiv: 0810.3112
MSC: 34M35; 33E10; 34M55
Received: November 26, 2008; in final form March 25, 2009; Published online April 3, 2009
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Citation: Kouichi Takemura, “Middle Convolution and Heun's Equation”, SIGMA, 5 (2009), 040, 22 pp.

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Simon N. M. Ruijsenaars, “Hilbert–Schmidt Operators vs. Integrable Systems of Elliptic Calogero–Moser Type. III. The Heun Case”, SIGMA, 5 (2009), 049, 21 pp.
2. Filipuk G.V., “Middle convolution and the hypergeometric equation”, J. Phys. A, 43:17 (2010), 175204, 10 pp.
3. Takemura K., “Integral transformation and Darboux transformation of Heun's differential equation”, Nonlinear and Modern Mathematical Physics, Proceedings of the First International Workshop, AIP Conference Proceedings, 1212, 2010, 58–65
4. Takemura K., “Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial”, Journal of Physics A-Mathematical and Theoretical, 45:8 (2012), 085211
5. Takemura K., “Integral Transformation of Heun's Equation and Apparent Singularity”, Painleve Equations and Related Topics (2012), Degruyter Proceedings in Mathematics, eds. Bruno A., Batkhin A., Walter de Gruyter & Co, 2012, 257–261
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
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
8. Yulia Bibilo, Galina Filipuk, “Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution”, SIGMA, 11 (2015), 023, 14 pp.
9. Bibilo Yu. Filipuk G., “Middle Convolution and Non-Schlesinger Deformations”, Proc. Jpn. Acad. Ser. A-Math. Sci., 91:5 (2015), 66–69
10. 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
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