### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 053, 33 pages      arXiv:2009.07318      https://doi.org/10.3842/SIGMA.2021.053

### Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind

Howard S. Cohl a, Justin Park b and Hans Volkmer c
a) Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, CA 92694, USA
b) Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
c) Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53211, USA

Received September 17, 2020, in final form May 04, 2021; Published online May 20, 2021

Abstract
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 which 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 $[1,\infty)$. In order 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.

Key words: Ferrers functions; associated Legendre functions; Gauss hypergeometric function.

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