Heine's identity

In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine is a Fourier expansion of a reciprocal square root which Heine presented as

${\frac {1}{\sqrt {z-\cos \psi }}}={\frac {\sqrt {2}}{\pi }}\sum _{m=-\infty }^{\infty }Q_{m-{\frac {1}{2}}}(z)e^{im\psi }$

where $Q_{m-{\frac {1}{2}}}$ is a Legendre function of the second kind, which has degree, m − ¹/₂, a half-integer, and argument, z, real and greater than one. This expression can be generalized for arbitrary half-integer powers as follows

$(z-\cos \psi )^{n-{\frac {1}{2}}}={\sqrt {\frac {2}{\pi }}}{\frac {(z^{2}-1)^{\frac {n}{2}}}{\Gamma ({\frac {1}{2}}-n)}}\sum _{m=-\infty }^{\infty }{\frac {\Gamma (m-n+{\frac {1}{2}})}{\Gamma (m+n+{\frac {1}{2}})}}Q_{m-{\frac {1}{2}}}^{n}(z)e^{im\psi },$

where $\scriptstyle \,\Gamma$ is the Gamma function.