Аннотация:
In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I–VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonlogarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlevé equations. In particular, Painlevé equations can be also divided into 5 supertypes, and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the “time variable” may depend on the type. We describe unified treatments for several possible “time variables”.
Ключевые слова:
linear ordinary differential equation, Heun class equations, isomonodromy deformations, Painlevé equations.
A.I. acknowledges the support by the Armenian Science Committee (SC Grant No. 20RF-171), and the Armenian National Science and Education Fund (ANSEF Grant No. PS5701). The work of J.D. and A.L. was supported by National Science Center (Poland) under the grant UMO-2019/35/B/ST1/01651.
Поступила:25 августа 2020 г.; в окончательном варианте 25 мая 2021 г.; опубликована 7 июня 2021 г.
Образец цитирования:
Jan Dereziński, Artur Ishkhanyan, Adam Latosiński, “From Heun Class Equations to Painlevé Equations”, SIGMA, 17 (2021), 056, 59 pp.
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Эта публикация цитируется в следующих 4 статьяx:
Mengkun Zhu, Yuting Chen, Jianduo Yu, Chuanzhong Li, “Orthogonal polynomials: From Heun equations to Painlevé equations”, Journal of Mathematical Physics, 66:1 (2025)
С. И. Тертычный, “О сохраняющей монодромию деформации дважды конфлюэнтного уравнения Гойна”, Топология, геометрия, комбинаторика и математическая физика, Сборник статей. К 80-летию члена-корреспондента РАН Виктора Матвеевича Бухштабера, Труды МИАН, 326, МИАН, М., 2024, 330–367; S. I. Tertichniy, “On the Monodromy-Preserving Deformation of a Double Confluent Heun Equation”, Proc. Steklov Inst. Math., 326 (2024), 303–338
Alfred Michel Grundland, Danilo Latini, Ian Marquette, “Recurrence Relations and General Solution of the Exceptional Hermite Equation”, Ann. Henri Poincaré, 2023
V. Chalifour, A. M. Grundland, “General solution of the exceptional Hermite differential equation and its minimal surface representation”, Ann. Henri Poincaré, 21:10 (2020), 3341
Цитирование в формате AMSBIB
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