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Generalized Heine’s identity for complex Fourier series of binomials

In his treatise, Heine (Heine 1881 In Theorie und Anwendungen) gave an identity for the Fourier series of the function , with , and z>1, in terms of associated Legendre functions of the second kind . In this paper, we generalize Heine’s identity for the function , with , and , in terms of . We also compute closed-form expressions for some .