Heine's identity

In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine [cite book
last = Heine
first = Heinrich Eduard
title = Handbuch der Kugelfunctionen, Theorie und Andwendungen
publisher = Physica-Verlag
date = 1881
place = Wuerzburg "(See [http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=01410002&seq=&view=50&frames=0&pagenum=286 page 286] )"] is a Fourier expansion of a reciprocal square root which Heine presented as

:$frac\{1\}\{sqrt\{z-cospsi=frac\{sqrt\{2\{pi\}sum\_\{m=-infty\}^infty\; Q\_\{m-frac12\}(z)\; e^\{impsi\}$

where [cite journal | last=Cohl | first=Howard S. |coauthors=J.E. Tohline, A.R.P. Rau;H.M. Srivastava | title=Developments in determining the gravitational potential using toroidal functions | year=2000 | journal=Astronomische Nachrichten | issn=0004-6337 | volume=321 | issue=5/6 | pages=363–372 | doi=10.1002/1521-3994(200012)321:5/6<363::AID-ASNA363>3.0.CO;2-X | doilabel=10.1002/1521-3994(200012)321:5/6363::AID-ASNA3633.0.CO;2-X] $Q\_\{m-frac12\}$ 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 [cite conference
first = H. S.
last = Cohl
title = Portent of Heine's Reciprocal Square Root Identity
booktitle = 3D Stellar Evolution, ASP Conference Proceedings, held 22-26 July 2002 at University of California Davis, Livermore, California, USA. Edited by Sylvain Turcotte, Stefan C. Keller and Robert M. Cavallo
volume = 293
isbn = 1583811400
year = 2003 ] for arbitrary half-integer powers as follows

:$(z-cospsi)^\{n-frac12\}=sqrt\{frac\{2\}\{pifrac\{(z^2-1)^\{frac\{n\}\{2\}\{Gamma(frac12-n)\}sum\_\{m=-infty\}^\{infty\}frac\{Gamma(m-n+frac12)\}\{Gamma(m+n+frac12)\}Q\_\{m-frac12\}^n(z)e^\{impsi\},$

where $scriptstyle,Gamma$ is the Gamma function.

References