\begin{equation}
R^S \propto \left\{ \begin{array}{ll}
\sqrt{\left(2l+3\right)}R^{\alpha}_{l+1}-2\sqrt{\left(2l+1\right)}R^{\alpha}_{l-1} & \mbox{if $\kappa^L = l$};\\
R^{\alpha}_{l+1} & \mbox{if $\kappa^L = -\left(l+1\right)$}.\end{array} \right.


\end{equation}