Generalized Heine's identity for complex Fourier series of binomials

Abstract

In his treatise, Heine (Heine 1881 In Theorie und Anwendungen) gave an identity for the Fourier series of the function (z - cos ψ)-1/2, with z, ψ ε ℝ, and z > 1, in terms of associated Legendre functions of the second kind Qn-1/20(z). In this paper, we generalize Heine's identity for the function (z - cos ψ)-μ, with μ ε ℂ, ψ ε ℝ and z ε ℂ \ (-∞, 1], in terms of Qn-1/2μ-1/2(z). We also compute closed-form expressions for some Qnμ (z). © 2010 The Royal Society.