\[{H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim e^{\pm\mathrm{i}{\theta_{\ell}}%
\left(\eta,\rho\right)}\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{\left(b%
\right)_{k}}}{k!(\pm 2\mathrm{i}\rho)^{k}},\]