The Teukolsky-Starobinsky identites relate solutions of the two Newman-Penrose scalars $\Psi_0$ and $\Psi_4$ around Kerr black holes. For example (here I am quoting Theorem 1 in Sec 81 of Chandrasekhar's book on black holes):
\begin{equation} \Delta^2D_0^+D_0^+D_0^+D_0^+\Delta^2R_{+2}\propto R_{-2}, \end{equation}
where $R_+$ and $R_-$ are related to $\Psi_0$ and $\Psi_4$, respectively, and the terms $\Delta$ and $D_0^+$ are derivative operators that depend on the Kerr geometry. For more context, the above relation is assuming we write, e.g.
\begin{equation} \Psi_0(t,r,\theta,\phi)=e^{i\omega t+im\phi}S(\theta)R_+(r). \end{equation}
My question is: are there versions of the Teukolsky-Starobinsky identities written out in terms of the Newman-Penrose scalars and derivative operators? What I would like is an expression of the form
\begin{equation} \Delta \delta etc \times \Psi_4 = \Psi_0, \end{equation}
instead of expressions in terms of the $R$ functions. Either an answer here or a link to references would both be helpful!