Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind
We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2021 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2021
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211296 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind. Howard S. Cohl, Justin Park and Hans Volkmer. SIGMA 17 (2021), 053, 33 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862533145340936192 |
|---|---|
| author | Cohl, Howard S. Park, Justin Volkmer, Hans |
| author_facet | Cohl, Howard S. Park, Justin Volkmer, Hans |
| citation_txt | Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind. Howard S. Cohl, Justin Park and Hans Volkmer. SIGMA 17 (2021), 053, 33 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the 18 hypergeometric arguments that correspond to these representations, we give geometrical descriptions of the corresponding convergence regions in the complex plane. In addition, we consider a corresponding single sum Fourier expansion for the Ferrers function of the second kind. In four of the eighteen cases, the determination of the Ferrers function of the second kind requires the evaluation of the hypergeometric function separately above and below the branch cut at [1,∞). To complete these derivations, we use well-known results to derive expressions for the hypergeometric function above and below its branch cut. Finally, we give a detailed review of the 1888 paper by Richard Olbricht, who was the first to study hypergeometric representations of Legendre functions.
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| first_indexed | 2026-03-12T14:20:56Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211296 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T14:20:56Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cohl, Howard S. Park, Justin Volkmer, Hans 2025-12-29T11:04:39Z 2021 Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind. Howard S. Cohl, Justin Park and Hans Volkmer. SIGMA 17 (2021), 053, 33 pages 1815-0659 2020 Mathematics Subject Classification: 33C05; 33C55; 42B05 arXiv:2009.07318 https://nasplib.isofts.kiev.ua/handle/123456789/211296 https://doi.org/10.3842/SIGMA.2021.053 We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the 18 hypergeometric arguments that correspond to these representations, we give geometrical descriptions of the corresponding convergence regions in the complex plane. In addition, we consider a corresponding single sum Fourier expansion for the Ferrers function of the second kind. In four of the eighteen cases, the determination of the Ferrers function of the second kind requires the evaluation of the hypergeometric function separately above and below the branch cut at [1,∞). To complete these derivations, we use well-known results to derive expressions for the hypergeometric function above and below its branch cut. Finally, we give a detailed review of the 1888 paper by Richard Olbricht, who was the first to study hypergeometric representations of Legendre functions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind Article published earlier |
| spellingShingle | Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind Cohl, Howard S. Park, Justin Volkmer, Hans |
| title | Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind |
| title_full | Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind |
| title_fullStr | Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind |
| title_full_unstemmed | Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind |
| title_short | Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind |
| title_sort | gauss hypergeometric representations of the ferrers function of the second kind |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211296 |
| work_keys_str_mv | AT cohlhowards gausshypergeometricrepresentationsoftheferrersfunctionofthesecondkind AT parkjustin gausshypergeometricrepresentationsoftheferrersfunctionofthesecondkind AT volkmerhans gausshypergeometricrepresentationsoftheferrersfunctionofthesecondkind |