Generalized Heine Identity for Complex Fourier Series of Binomials

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 \(1/[z-\cos\psi]^{1/2}\), for \(z,\psi\in\R\), 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 \(1/[z-\cos\psi]^\mu\), 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 \(\frac12-\mu\). We are also able to compute certain closed-form expressions for associated Legendre functions of the second kind.