Heine's identity

In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine[1] is a Fourier expansion of a reciprocal square root which Heine presented as

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

where[2] ${\displaystyle Q_{m-{\frac {1}{2}}}}$ 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[3] for arbitrary half-integer powers as follows

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

where ${\displaystyle \scriptstyle \,\Gamma }$ is the Gamma function.