\[\int z\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(az)\,\mathrm{d}z=%
\tfrac{1}{4}z^{2}\left(2\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(az)-%
\mathscr{C}_{\mu-1}\left(az\right)\mathscr{D}_{\mu+1}(az)-\mathscr{C}_{\mu+1}%
\left(az\right)\mathscr{D}_{\mu-1}(az)\right),\]