The confluent Heun equation with an added apparent singular point is under consideration. A new integral transform connecting solutions of this equation with different parameters is obtained. The kernel of this transform is a suitable solution of the confluent hypergeometric equation. Bibliography: 22 titles.
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References
H. Bateman and A. Erdelyi, Higher Transcendental Functions, V. 1, McCraw-Hill Book Company Inc., New York (1953).
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards (1964).
W. Wasow, Asymptotic Expansions For Ordinary Differential Equations, John Wiley, New York (1965).
A. A. Bolibrukh, Fuchsian Differential Equations and Holomorphic Bundles [in Russian], MCCME, Moscow (2000).
Y. Sibuya, Linear ODE in Complex Domain. Analytic Continuation, AMS, Providence, Rhode Island (1990).
K. Iwasaki, H. Kimura, S. Shimomura, and M. Yosida, From Gauss to Painlevé: A Modern Theory of Special Functions, Braunshweig, Vieweg (1991).
S. Yu. Slavyanov and W. Lay, Special Functions: A Unified Theory Based on Singularities, Oxford University Press, Oxford, New York (2000).
A. Ya. Kazakov, “Euler integral symmetry and deformed hypergeometric equation,” J. Math. Sciences, 185, No. 4, 573–580 (2012).
A. Ya. Kazakov, “Monodromy of Heun equations with apparent singularities,” Intern. Journ. Theor. Math. Phys., 3, No. 6, 190–196 (2013).
A. V. Shanin and R. V. Craster, “Removing false singular points as a method of solving ordinary differential equations,” Euro. J. Appl. Math., 13, 617–639 (2002).
A. Ishkhanyan and K. A. Suominen, “New solutions of Heun’s general equation,” J. Phys. A, 36, L81–L85 (2003).
A. Ya. Kazakov, “Integral symmetry, integral invariants, and monodromy of ordinary differential equations,” Theor. Math. Phys., 116, No. 3, 991–1000 (1998).
A. Ya. Kazakov and S. Yu. Slavyanov, “Integral relations for the special functions of the Heun class,” Theor. Math. Phys., 107, No. 3, 388–396 (1996).
L. J. El-Jaick and B. D. B. Figueiredo, “Transformation of Heun equation and its integral relations,” J. Phys. A., 44, 075204 (2011).
A. Ya. Kazakov and S. Yu. Slavyanov, “Euler integral symmetries for a deformed Heun equation and symmetries of the Painlev´e PVI equation,” Theor. Math. Phys., 155, No. 2, 721–732 (2008).
K. Takemura, “Middle convolution and Heun’s equation,” SIGMA, 5, 040 (2009).
M. Detweiler and S. Reiter, “Middle convolution of Fuchsian systems and the construction of rigid differential systems,” J. Algebra, 318, 1–24 (2007).
A. Ya. Kazakov and S. Yu. Slavyanov, “Euler integral symmetries for a deformed confluent Heun equation and symmetries of the Painlev´e PV equation,” Theor. Math. Phys., 179, 543–549 (2014).
A. Ya. Kazakov, “Symmetries of the confluent Heun equation,” J. Math. Sc., 117, No. 2, 3918–3927 (2003).
A. Ya. Kazakov, “Isomonodromy deformation of the Heun class equation,” in: Painlevé Equations and Related Topics, ed. by A. D. Bruno, A. B. Batkhin, De Gruyter (2012), pp. 107–116.
A. Castro, J. M. Lapan, A. Maloney, and M. J. Rodriguez, “Black hole scattering from monodromy,” Class. Quant. Gravity, 30, 165005 (2013).
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Inc (1994).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 426, 2014, pp. 34–48.
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Kazakov, A.Y. Integral Symmetry for the Confluent Heun Equation with an Added Apparent Singularity. J Math Sci 214, 268–276 (2016). https://doi.org/10.1007/s10958-016-2776-3
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DOI: https://doi.org/10.1007/s10958-016-2776-3