Generalized Heine Identity for Complex Fourier Series of Binomials

TL;DR

This paper extends Heine's classical Fourier series identity for specific functions to a more general form involving complex parameters, providing new formulas and closed-form expressions for associated Legendre functions.

Contribution

It generalizes Heine's identity to complex powers and parameters, offering new Fourier series representations and closed-form formulas for associated Legendre functions.

Findings

Generalized Fourier series for $1/[z- ext{cos}\psi]^\mu$ with complex parameters.

Derived closed-form expressions for associated Legendre functions of the second kind.

Extended classical identities to broader complex domains.

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 $z,\psi\in\R$, 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.