Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind

We derive all eighteen Gauss hypergeometric representations for the Ferrers function of second kind, each with a different argument. They are obtained from associated Legendre kind by using limit representation. For 18 arguments which correspond to these representations, we give geometrical descriptions corresponding convergence regions in complex plane. In addition, consider single sum Fourier expansion kind. four cases, determination requires evaluation separately above and below branch cut at [1,infty). order complete derivations, use well-known results expressions its cut. Finally detailed review 1888 paper Richard Olbricht who was first study functions.