Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2021
• Main Authors: Cohl, Howard S., Park, Justin, Volkmer, Hans
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2021
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211296
• 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

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.