Generalized Heine Identity for Complex Fourier Series of Binomials

Abstract

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 $z,\psi \in \mathbb{R}$, 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 $z,μ\in\C$, $|z|>1$, and the coefficients of the expansion are given in terms of the same functions with order given by $\frac{1}{2}-\mu$. We are also able to compute certain closed-form expressions for associated Legendre functions of the second kind.

12 pages

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