Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind

TL;DR

This paper systematically derives all eighteen Gauss hypergeometric representations of the Ferrers function of the second kind, providing geometric insights into their convergence regions and addressing branch cut evaluations, thus enriching the mathematical understanding of these special functions.

Contribution

It presents a complete set of hypergeometric representations for the Ferrers function of the second kind, including new geometric descriptions and branch cut evaluations, building on historical work.

Findings

Eighteen hypergeometric representations derived

Geometrical descriptions of convergence regions provided

Explicit evaluations near branch cuts included

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.