Ferrers Functions of Arbitrary Degree and Order and Related Functions

Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers functions of different orders and degrees as well as a uniform asymptotic expansion are derived in this article. Simple proofs of four generating functions for Ferrers functions are given. Addition theorems for P \(_{\nu }^{-\mu }\left( \tanh \left( \alpha +\beta \right) \right) \) ${}_{\nu }^{-\mu }\left(tanh,\left(\alpha ,+,\beta \right)\right)$ are proved by basing on generation functions for three families of hypergeometric polynomials. Relations for Gegenbauer polynomials and Ferrers associated Legendre functions (associated Legendre polynomials) are obtained as special cases.