Rewriting the transition amplitude, second step

We have $$ O_{\mu}^{\lambda} = \sum_{pq} \langle p \vert \vert O^{\lambda}\vert \vert q \rangle (-1)^{j_p-m_p}\left(\begin{array}{ccc} j_p & \lambda & j_q \\ -m_p & \mu & m_q\end{array}\right)a^{\dagger}_pa_q. $$ We then single out the sum over \( m_p \) and \( m_q \) only and define the recoupled one-body part of the operator as $$ \lambda^{-1}\left[ a_{j_p}^{\dagger}\tilde{a}_{j_q}\right]^{\lambda}_{\mu} = \sum_{m_p,m_q} (-1)^{j_p-m_p}\left(\begin{array}{ccc} j_p & \lambda & j_q \\ -m_p & \mu & m_q\end{array}\right)a^{\dagger}_pa_q, $$ with \( \lambda=\sqrt{2\lambda+1} \). This gives the following expression for the one-body operator $$ O_{\mu}^{\lambda} = \sum_{j_pj_q} \langle p \vert \vert O^{\lambda}\vert \vert q \rangle \lambda^{-1}\left[ a_{j_p}^{\dagger}\tilde{a}_{j_q}\right]^{\lambda}_{\mu}. $$