Geodesic
Representations and use of symbolic computations in the theory of Heun equations
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 162-176 
A. Ya. Kazakov; S. Yu. Slavyanov. Representations and use of symbolic computations in the theory of Heun equations. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 162-176. http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a9/
@article{ZNSL_2015_432_a9,
     author = {A. Ya. Kazakov and S. Yu. Slavyanov},
     title = {Representations and use of symbolic computations in the theory of {Heun} equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {162--176},
     year = {2015},
     volume = {432},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a9/}
}
TY  - JOUR
AU  - A. Ya. Kazakov
AU  - S. Yu. Slavyanov
TI  - Representations and use of symbolic computations in the theory of Heun equations
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2015
SP  - 162
EP  - 176
VL  - 432
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a9/
LA  - en
ID  - ZNSL_2015_432_a9
ER  - 
%0 Journal Article
%A A. Ya. Kazakov
%A S. Yu. Slavyanov
%T Representations and use of symbolic computations in the theory of Heun equations
%J Zapiski Nauchnykh Seminarov POMI
%D 2015
%P 162-176
%V 432
%U http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a9/
%G en
%F ZNSL_2015_432_a9

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A first-order $2\times2$ system equivalent to the Heun equation is obtained. A deformed Heun equation in symmetric form is presented. Series solutions of this equation are presented. A four-parameter subfamily of deformed confluent Heun equation whose solutions have integral representations is found.

[1] S. Yu. Slavyanov, W. Lay, Special Functions: A Unified Theory Based on Singularities, Oxford Univ. Press, 2000 | MR | Zbl

[2] A. A. Bolibrukh, Inverse Monodromie Problems in Analytical Theory of Differential Equations, MCCME, Moscow, 2009 (in Russian)

[3] M. V. Babich, “About a canonical parametrization of the phase spaces of equations of isomonodromic deformations of Fuchs systems of dimension $2\times2$”, Russian Math. Surv., 64:1 (2009), 45–127 | DOI | MR | Zbl

[4] A. Ya. Kazakov, S. Yu. Slavyanov, “Euler integral symmetries for a deformed symmetries of the Painleve PVI equation”, Theor. Math. Phys., 155:2 (2008), 722–733 | DOI | MR | Zbl

[5] S. Yu. Slavyanov, F. R. Vukajlovic, “Isomonodromic deformations and “antiquantization” for the simplest ordinary differential equations”, Theor. Math. Phys., 150:1 (2007), 123–131 | DOI | MR | Zbl

[6] A. Ya. Kazakov, “Euler integral symmetry and deformed hypergeometric equation”, J. Math. Sci., 185:4 (2012), 573–580 | DOI | MR | Zbl

[7] A. Ya. Kazakov, “Monodromy of Heun equations with apparent singularities”, Intern. J. Theor. Math. Phys., 3:6 (2013), 190–196

[8] A. V. Shanin, R. V. Craster, “Removing false singular points as a method of solving ordinary differential equations”, Euro. J. Appl. Math., 13 (2002), 617–639 | DOI | MR | Zbl

[9] A. Ishkhanyan, K. A. Suominen, “New solutions of Heun's general equation”, J. Phys. A, 36 (2003), L81–L85 | DOI | MR | Zbl

[10] A. Ya. Kazakov, S. Yu. Slavyanov, “Euler integral symmetries for a deformed confluent Heun equation and symmetries of the Painleve PV equation”, Theor. Math. Phys., 179:2 (2014), 543–549 | DOI | Zbl

[11] A. Ya. Kazakov, “Integral symmetry of the confluent Heun equation with added apparent singularity”, Zap. Nauchn. Semin. POMI, 426, 2014, 34–48

[12] A. Ya. Kazakov, “Isomonodromy deformation of the Heun class equation”, Painleve Equations and related topics, De Gruyter Proceedings in mathematics, eds. A. D. Bruno, A. B. Batkhin, 2012, 107–116 | MR | Zbl

[13] A. Ya. Kazakov, “Integral symmetry, integral invariants and monodromy of ordinary differential equations”, Theor. Math. Phys., 116:3 (1998), 991–1000 | DOI | MR | Zbl

[14] A. Ya. Kazakov, “Symmetries of the confluent Heun equation”, J. Math. Sci., 117:2 (2003), 3918–3927 | DOI | MR | Zbl

[15] T. Oshima, Fractional calculus of Weyl algebra and fuchsian differential equations, Preprint | MR

[16] R. S. Maier, “On reducing the Heun equation to the hypergeometric equation”, J. Diff. Equat., 213 (2005), 171–203 | DOI | MR | Zbl

[17] A. Ronveaux, “Factorization of the Heun's differential operator”, Appl. Math. Comp., 141:1 (2003), 177–184 | DOI | MR | Zbl