The solid harmonic operator functions \( \mathcal {Y}_{l 0}({\bf J})\)  in Eq.(REF ) were defined [1]} by the generating function
[1]}, [3]}, [4]}, [5]}
\(& &\frac{1}{2^{l}l!} \left[ \frac{2l+1}{4\pi } \right]^{\!1/2} \left[-z_{+}^2(J_x +iJ_y) + z_{-}^2(J_x-iJ_y)+2z_{+}z_{-}J_z \right]^{l}\nonumber \\& = & \sum _{m=-l}^{l} \; \frac{ z_{+}^{l+m}\; z_{-}^{l-m} }{ \left[(l+m)!(l-m)!\right]^{1/2}} \;\mathcal {Y}_{lm}({\bf J}) \) 

Операторные функции твёрдого гармоника \( \mathcal {Y}_{l 0}({\bf J})\) в уравнении (REF ) были определены [1] с помощью порождающей функции

\(& &\frac{1}{2^{l}l!} \left[ \frac{2l+1}{4\pi } \right]^{\!1/2} \left[-z_{+}^2(J_x +iJ_y) + z_{-}^2(J_x-iJ_y)+2z_{+}z_{-}J_z \right]^{l}\nonumber \\& = & \sum _{m=-l}^{l} \; \frac{ z_{+}^{l+m}\; z_{-}^{l-m} }{ \left[(l+m)!(l-m)!\right]^{1/2}} \;\mathcal {Y}_{lm}({\bf J}) \)