In this paper we generalize an identity first given by Heinrich Eduard Heine
in his treatise, {\it Handbuch der Kugelfunctionen, Theorie und Anwendungen
(1881), which gives a Fourier series for 1/[z−cosψ]1/2, for
z,ψ∈R, and z>1, in terms of associated Legendre functions of the
second kind with odd-half-integer degree and vanishing order. In this paper we
give a generalization of this identity as a Fourier series of
1/[z−cosψ]μ, where z,\mu\in\C, ∣z∣>1, and the coefficients of the
expansion are given in terms of the same functions with order given by
21−μ. We are also able to compute certain closed-form expressions for
associated Legendre functions of the second kind.Comment: 12 page