\[F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\rho^{\ell+1}e^{\mp%
\mathrm{i}\rho}M\left(\ell+1\mp\mathrm{i}\eta,2\ell+2,\pm 2\mathrm{i}\rho%
\right),\]