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arXiv · 0912.0126

Generalized Heine Identity for Complex Fourier Series of Binomials

Howard S. Cohl·Diego E. Dominici
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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 $1/[z-\cosψ]^{1/2}$, for $z,ψ\in\R$, 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 $1/[z-\cosψ]^μ$, 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 $\frac12-μ$. We are also able to compute certain closed-form expressions for associated Legendre functions of the second kind.

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Howard S. Cohl, Diego E. Dominici. 2009-12-01. Generalized Heine Identity for Complex Fourier Series of Binomials. https://arxiv.org/abs/0912.0126

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