Geodesic
Euler integral symmetries and the asymptotic of the monodromy for the Heun equation
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 50, Tome 493 (2020), pp. 186-199 Cet article a éte moissonné depuis la source Math-Net.Ru

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Euler integral transform connects monodromy matrices of the Heun equation with different sets of parameters. In this paper, this fact is used to calculate the asymptotic behavior of the monodromy of the Heun confluent equation in the case of the presence of a “combined” singularity. 
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A. Ya. Kazakov. Euler integral symmetries and the asymptotic of the monodromy for the Heun equation. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 50, Tome 493 (2020), pp. 186-199. http://geodesic.mathdoc.fr/item/ZNSL_2020_493_a12/

[1] A. Ronveaux (ed.), Heun's differential equations, Oxford University Press, Oxford–New York–Tokyo, 1995 | Zbl

[2] S. Yu. Slavyanov, W. Lay, Special functions. A unified theory based on singularities, Oxford University Press, Oxford, 2000 | MR | Zbl

[3] I. V. Komarov, L. I. Ponomarev, S. Yu. Slavyanov, Sferoidalnye i kulonovskie sferoidalnye funktsii, Nauka, M., 1976

[4] H. Suzuki, E. Takasugi, H. Umetsu, “Perturbations of Kerr-de Sitter Black Holes and Heun's Equations”, Progress Theor. Phys., 100:3 (1998), 491–505 | DOI | MR

[5] D. Vincenzi, E.Bodenschatz, “Single polymer dynamics in elongational flow and the confluent Heun equation”, J. Phys. A, 39 (2006), 10691–10701 | DOI | MR | Zbl

[6] A. Castro, J. M. Lapan, A. Maloney, M. J. Rodriguez, “Black Hole Scattering from Monodromy”, Classical and Quantum Gravity, 30:16 (2013) | DOI | MR

[7] P. Fiziev, D. Staicova, “Application of the confluent Heun functions for finding the quasinormal modes of nonrotating black holes”, Phys. Rev. D, 84 (2011), 127502 | DOI

[8] M. S. Cunha, H. R. Christiansen, “Confluent Heun functions in gauge theories on thick braneworlds”, Phys. Rev. D, 84 (2011), 085002 | DOI

[9] D. Staicova, P. Fiziev, “The Heun Functions and Their Applications in Astrophysics”, International Workshop on Lie Theory and Its Applications in Physics (Varna, Bulgaria, June, 2015), ed. V. Dobrev, 303–308 | MR | Zbl

[10] A. M. Ishkanyan, “Schrödinger potentials solvable in terms of the confluent Heun functions”, Theor. Math. Phys., 188 (2016), 980–993 | DOI | MR

[11] H. S. Vieira, V. B. Bezerra, “Confluent Heun functions and the physics of black holes: resonant frequencies, Hawking radiation and scattering of scalar waves”, Annals of Physics, 2016 | DOI | MR

[12] T. Birkandan, M. Hortacsu, “Quantum Field Theory Applications of Heun Type Functions”, Rept. Math. Phys., 79:81 (2017), arXiv: 1605.07848 [hep-th] | MR | Zbl

[13] L. Andersson, S. Ma, C. Paganini, B. F. Whiting, “Mode stability on the real axis”, J. Math. Phys., 58 (2017), 072501 | DOI | MR | Zbl

[14] M. Bednarik, M. Cervenka, “Description of waves in inhomogeneous domains using Heun's equation”, Waves in Random and Complex Media, 28:2 (2018), 236–252 | DOI | MR

[15] M. Hortaçsu, “Heun Functions and Some of Their Applications in Physics”, Advances in High Energy Physics, 2018, 8621573 | Zbl

[16] O. V. Motygin, “On numerical evaluation of the Heun functions”, Proceedings of Days on Diffraction, 2015, 222–227, arXiv: 1506.03848 [math.NA]

[17] O. V. Motygin, On evaluation of the confluent Heun functions, 2018, arXiv: 1804.01007v1 [math.NA]

[18] A. Ya. Kazakov, S. Yu. Slavyanov, “Integral relations for the Heun-class special functions”, Theor. Math. Phys., 107 (1996), 733–738 | DOI | MR

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

[20] A. Ya. Kazakov, S. Yu. Slavyanov, “Euler integral symmetries for a deformed Heun equation and symmetries of the Painleve VI equation”, Theor. Math. Phys., 155:2 (2008), 721–732 | DOI | MR

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

[22] E. Hille, Ordinary Differential Equations in the Complex Domain, Dover Publications Inc., New York, 1997 | MR | Zbl

[23] T. M. Cherry, “Uniform asymptotic formulae for functions with transition points”, Transactions of the American Mathematical Society, 68:1 (1950), 224–257 | DOI | MR | Zbl

[24] F. W. J .Olver, Asymptotics and special functions, Academic Press, New York–San-Francisco–London, 1974 | MR

[25] S. Slavyanov, Asymptotic solutions of the one-dimensional Schrödinger equation, Translations of mathematical monographs, 151, American Mathematical Society, Providence, 1996, xvi+190 pp. | DOI | MR

[26] V. M. Babich, K. D. Cherednichenko, “On a differential equation with a singular point of regular type and a large parameter”, Integral Transforms and Special Functions, 11:2 (2001), 101–112 | DOI | MR | Zbl

[27] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions, National Bureu of Standards, Washington, 1964 | MR