Английская Википедия:Heine's identity

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Шаблон:Short description 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 <math display="block">\frac{1}{\sqrt{z-\cos\psi}} = \frac{\sqrt{2}}{\pi}\sum_{m=-\infty}^\infty Q_{m-\frac12}(z) e^{im\psi}</math> where[2] <math> Q_{m-\frac12}</math> is a Legendre function of the second kind, which has degree, m − Шаблон: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 <math display="block">(z-\cos\psi)^{n-\frac12} = \sqrt{\frac{2}{\pi}}\frac{(z^2-1)^{\frac{n}{2}}}{\Gamma(\frac12-n)} \sum_{m=-\infty}^{\infty} \frac{\Gamma(m-n+\frac12)}{\Gamma(m+n+\frac12)}Q_{m-\frac12}^n(z)e^{im\psi},</math> where <math>\scriptstyle\,\Gamma</math> is the Gamma function.

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