From Heun Class Equations to Painlevé Equations
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...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2021
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211367 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | From Heun Class Equations to Painlevé Equations. Jan Dereziński, Artur Ishkhanyan and Adam Latosiński. SIGMA 17 (2021), 056, 59 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862640366125056000 |
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| author | Dereziński, Jan Ishkhanyan, Artur Latosiński, Adam |
| author_facet | Dereziński, Jan Ishkhanyan, Artur Latosiński, Adam |
| citation_txt | From Heun Class Equations to Painlevé Equations. Jan Dereziński, Artur Ishkhanyan and Adam Latosiński. SIGMA 17 (2021), 056, 59 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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's 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 non-logarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlevé equations. In particular, Painlevé equations can also be 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''.
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| first_indexed | 2026-03-15T05:48:35Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211367 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T05:48:35Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Dereziński, Jan Ishkhanyan, Artur Latosiński, Adam 2025-12-30T15:57:32Z 2021 From Heun Class Equations to Painlevé Equations. Jan Dereziński, Artur Ishkhanyan and Adam Latosiński. SIGMA 17 (2021), 056, 59 pages 1815-0659 2020 Mathematics Subject Classification: 34A30; 34B30; 34M55; 34M56 arXiv:2007.05698 https://nasplib.isofts.kiev.ua/handle/123456789/211367 https://doi.org/10.3842/SIGMA.2021.056 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's 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 non-logarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlevé equations. In particular, Painlevé equations can also be 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''. J.D. and A.I. would like to express their gratitude to Galina Filipuk for very useful discussions and remarks. 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 the National Science Center (Poland) under the grant UMO-2019/35/B/ST1/01651. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications From Heun Class Equations to Painlevé Equations Article published earlier |
| spellingShingle | From Heun Class Equations to Painlevé Equations Dereziński, Jan Ishkhanyan, Artur Latosiński, Adam |
| title | From Heun Class Equations to Painlevé Equations |
| title_full | From Heun Class Equations to Painlevé Equations |
| title_fullStr | From Heun Class Equations to Painlevé Equations |
| title_full_unstemmed | From Heun Class Equations to Painlevé Equations |
| title_short | From Heun Class Equations to Painlevé Equations |
| title_sort | from heun class equations to painlevé equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211367 |
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