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 TMF, 1996, Volume 107, Number 3, Pages 388–396 (Mi tmf1164)

Integral relations for special functions of hypergeometric and Heun class

A. Ya. Kazakov, S. Yu. Slavyanov

Saint-Petersburg State University

Abstract: Ordinary differential equations with polynomial coefficients originate different kinds of integral relations for its solutions: integral representations in terms of simpler functions, integral equations etc. In this paper, a new kind of integral relations for functions of the Heun class are presented. These relations are coupling in ivolution eigensolutions, which are characterized by different behaviour at singularities and often also by different intervals of consideration and equations themselves. The studied relations are arranged in two staircases where each succeeding equation may be obtained with the help of the confluence process.

DOI: https://doi.org/10.4213/tmf1164

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English version:
Theoretical and Mathematical Physics, 1996, 107:3, 733–739

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Citation: A. Ya. Kazakov, S. Yu. Slavyanov, “Integral relations for special functions of hypergeometric and Heun class”, TMF, 107:3 (1996), 388–396; Theoret. and Math. Phys., 107:3 (1996), 733–739

Citation in format AMSBIB
\Bibitem{KazSla96} \by A.~Ya.~Kazakov, S.~Yu.~Slavyanov \paper Integral relations for special functions of hypergeometric and Heun class \jour TMF \yr 1996 \vol 107 \issue 3 \pages 388--396 \mathnet{http://mi.mathnet.ru/tmf1164} \crossref{https://doi.org/10.4213/tmf1164} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1407468} \zmath{https://zbmath.org/?q=an:0924.34021} \transl \jour Theoret. and Math. Phys. \yr 1996 \vol 107 \issue 3 \pages 733--739 \crossref{https://doi.org/10.1007/BF02070381} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1996WY91300004} 

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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. N. A. Veshev, “Degeneration of solutions of Heun's equation under fusion of singularities”, Theoret. and Math. Phys., 110:2 (1997), 179–182
2. Ochiai, H, “Non-commutative harmonic oscillators and Fuchsian ordinary differential operators”, Communications in Mathematical Physics, 217:2 (2001), 357
3. Tarasov, VF, “The Heun-Schrodinger radial equation with two auxiliary parameters for H-like atoms”, Modern Physics Letters B, 16:25 (2002), 937
4. D. P. Novikov, “Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation”, Theoret. and Math. Phys., 146:3 (2006), 295–303
5. Kouichi Takemura, “Middle Convolution and Heun's Equation”, SIGMA, 5 (2009), 040, 22 pp.
6. 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.
7. Takemura K., “Integral transformation and Darboux transformation of Heun's differential equation”, Nonlinear and Modern Mathematical Physics, AIP Conference Proceedings, 1212, 2010, 58–65
8. El-Jaick L.J., Figueiredo B.D.B., “Transformations of Heun's equation and its integral relations”, J. Phys. A: Math. Theor., 44:7 (2011), 075204
9. Takemura K., “Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial”, J. Phys. A: Math. Theor., 45:8 (2012), 085211
10. 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
11. A. Ya. Kazakov, “Integral symmetry for the confluent Heun equation with added apparent singularity”, J. Math. Sci. (N. Y.), 214:3 (2016), 268–276
12. Cordero R., Turrubiates F.J., Vera J.C., “On a Phase Space Quantum Description of the Spherical 2-Brane”, Phys. Scr., 89:7 (2014), 075001
13. El-Jaick L.J., Figueiredo B.D.B., “Integral Relations For Solutions of the Confluent Heun Equation”, Appl. Math. Comput., 256 (2015), 885–904
14. Takemura K., “Integral Transformation of Heun'S Equation and Some Applications”, J. Math. Soc. Jpn., 69:2 (2017), 849–891
15. Farrokh Atai, Edwin Langmann, “Series Solutions of the Non-Stationary Heun Equation”, SIGMA, 14 (2018), 011, 32 pp.
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