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~
Q_{-\frac{1}{2}}^\mu(z) + 2\sum_{n=1}^\infty Q^\mu_{n-\frac{1}{2}}(z) \cos(n\nu)

~=

~
e^{i\mu\pi}~\biggl(\frac{\pi}{2} \biggr)^{1 / 2} \Gamma(\mu + \tfrac{1}{2})\biggl[ \frac{(z^2-1)^{\mu/2}}{(z - \cos\nu)^{\mu + \frac{1}{2}}} \biggr]

A. Erdélyi (1953):  Volume I, §3.10, p. 166, eq. (3)

Valid for:    

~\mathrm{Re}~\mu > - \tfrac{1}{2}

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