Thoroidal function

$P_{\nu -1/2}^{\mu }(\func{ch}z)=\frac{2^{2\mu }}{\Gamma (1-\mu )}e^{-( +1/2)z}(1-e^{-2z})^{-\mu }{}_{2}F_{1}\left( \frac{1}{2}-\mu ,\nu -\mu -\frac{ 1}{2};1-2\mu ;1-e^{-2z}\right) , $$Q_{\nu -1/2}^{\mu }(\func{ch}z)=\sqrt{\pi }e^{i\mu \pi }\frac{\Gamma (\mu+\nu +\frac{1}{2})}{\Gamma (\nu +1)}e^{-(\nu +1/2)z}(1-e^{-2z})^{\mu}{}_{2}F_{1}\left( \frac{1}{2}-\mu ,\mu +\nu +\frac{1}{2};\mu+1;1-e^{-2z}\right) $

$P_{\nu -1/2}^{\mu }(\func{ch}z)=\frac{2^{2\mu }}{\Gamma (1-\mu )}e^{-( +1/2)z}(1-e^{-2z})^{-\mu }{}_{2}F_{1}\left( \frac{1}{2}-\mu ,\nu -\mu -\frac{ 1}{2};1-2\mu ;1-e^{-2z}\right) , $$Q_{\nu -1/2}^{\mu }(\func{ch}z)=\sqrt{\pi }e^{i\mu \pi }\frac{\Gamma (\mu+\nu +\frac{1}{2})}{\Gamma (\nu +1)}e^{-(\nu +1/2)z}(1-e^{-2z})^{\mu}{}_{2}F_{1}\left( \frac{1}{2}-\mu ,\mu +\nu +\frac{1}{2};\mu+1;1-e^{-2z}\right) $