\[{H^{\pm}_{\ell}}\left(\eta,\rho\right)=e^{\pm\mathrm{i}{\theta_{\ell}}\left(%
\eta,\rho\right)}(\mp 2\mathrm{i}\rho)^{\ell+1\pm\mathrm{i}\eta}U\left(\ell+1%
\pm\mathrm{i}\eta,2\ell+2,\mp 2\mathrm{i}\rho\right),\]