From these relations we have
$$\begin{aligned}& \mathcal{N}_{0,1}^{+}\mathcal{N}_{0,1}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta-1,\gamma,\delta,a,b-1 ) }(x,y,z) =n_{2} ( n_{2}+2n_{3}+\beta+\gamma+ \delta+b ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta-1,\gamma,\delta ,a,b-1 ) }(x,y,z), \\& \mathcal{N}_{0,1}\mathcal{N}_{0,1}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma,\delta,a,b ) }(x,y,z) = (n_{2}+1 ) ( n_{2}+2n_{3}+\beta+\gamma+ \delta+b+1 )P_{n_{1},n_{2},n_{3}}^{(\alpha,\beta,\gamma,\delta,a,b) }(x,y,z), \\& \mathcal{N}_{0,2}^{+}\mathcal{N}_{0,2}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta+1,\gamma,\delta,a,b-1 ) }(x,y,z) = (n_{2}+2n_{3}+\beta+\gamma+\delta+b+2 ) ( n_{2}+2n_{3}+\gamma+\delta+b+1 ) \\& \phantom{\mathcal{N}_{0,2}^{+}\mathcal{N}_{0,2}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta+1,\gamma,\delta,a,b-1 ) }(x,y,z) =}{}\times P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta+1,\gamma,\delta ,a,b-1 ) }(x,y,z), \\& \mathcal{N}_{0,2}\mathcal{N}_{0,2}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta+1,\gamma,\delta,a,b ) }(x,y,z) = (n_{2}+2n_{3}+\beta+\gamma+\delta+b+2 ) ( n_{2}+2n_{3}+\gamma+\delta+b+1 ) \\& \phantom{\mathcal{N}_{0,2}\mathcal{N}_{0,2}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta+1,\gamma,\delta,a,b ) }(x,y,z) =}{}\times P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta+1,\gamma,\delta ,a,b ) }(x,y,z), \\& \mathcal{N}_{0,3}^{+}\mathcal{N}_{0,3}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta-1,\gamma,\delta,a,b+1 ) }(x,y,z) = (n_{2}+\beta ) ( n_{2}+2n_{3}+\beta+ \gamma+\delta +b+2 ) \\& \phantom{\mathcal{N}_{0,3}^{+}\mathcal{N}_{0,3}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta-1,\gamma,\delta,a,b+1 ) }(x,y,z) =}{}\times P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta-1,\gamma ,\delta,a,b+1 )}(x,y,z), \\& \mathcal{N}_{0,3}\mathcal{N}_{0,3}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma,\delta,a,b+1 ) }(x,y,z) = (n_{2}+\beta ) ( n_{2}+2n_{3}+\beta+ \gamma+\delta +b+2 )\\& \phantom{\mathcal{N}_{0,3}\mathcal{N}_{0,3}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma,\delta,a,b+1 ) }(x,y,z) =}{}\times P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta,\gamma ,\delta,a,b+1 )}(x,y,z), \\& \mathcal{N}_{0,4}^{+}\mathcal{N}_{0,4}P_{n_{1},n_{2}-1,n_{3}}^{ (\alpha,\beta+1,\gamma,\delta,a,b ) }(x,y,z) =n_{2} ( n_{2}+\beta+1 ) P_{n_{1},n_{2}-1,n_{3}}^{ (\alpha,\beta +1,\gamma,\delta,a,b ) }(x,y,z), \\& \mathcal{N}_{0,4}\mathcal{N}_{0,4}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta+1,\gamma,\delta,a,b-1 ) }(x,y,z) =n_{2} ( n_{2}+\beta+1 ) P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta +1,\gamma,\delta,a,b-1 ) }(x,y,z), \\& \mathcal{N}_{0,5}^{+}\mathcal{N}_{0,5}P_{n_{1},n_{2}-1,n_{3}}^{ (\alpha,\beta,\gamma,\delta,a,b+1 ) }(x,y,z) =n_{2} ( n_{2}+2n_{3}+\gamma+\delta+b+2 ) P_{n_{1},n_{2}-1,n_{3}}^{ ( \alpha,\beta,\gamma,\delta,a,b+1 ) }(x,y,z), \\& \mathcal{N}_{0,5}\mathcal{N}_{0,5}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta-1,\gamma,\delta,a,b+1 ) }(x,y,z) =n_{2} ( n_{2}+2n_{3}+\gamma+\delta+b+2 ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta-1,\gamma,\delta,a,b+1 ) }(x,y,z), \\& \mathcal{N}_{0,6}^{+}\mathcal{N}_{0,6}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma,\delta,a,b-1 ) }(x,y,z) = (n_{2}+\beta ) ( n_{2}+2n_{3}+\gamma+ \delta+b+1 )P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta,\gamma,\delta ,a,b-1 )}(x,y,z), \\& \mathcal{N}_{0,6}\mathcal{N}_{0,6}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta-1,\gamma,\delta,a,b ) }(x,y,z) = (n_{2}+\beta ) ( n_{2}+2n_{3}+\gamma+ \delta+b+1 )P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta-1,\gamma,\delta ,a,b )}(x,y,z), \end{aligned}$$
and
$$\begin{aligned}& \mathcal{N}_{1,0}^{+}\mathcal{N}_{1,0}P_{n_{1},n_{2},n_{3}}^{ (\alpha-1,\beta,\gamma,\delta,a,b-1 ) }(x,y,z) =n_{1} ( n+n_{2}+n_{3}+e+1 ) P_{n_{1},n_{2},n_{3}}^{ (\alpha-1,\beta ,\gamma,\delta,a,b-1 ) }(x,y,z), \\& \mathcal{N}_{1,0}\mathcal{N}_{1,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma,\delta,a,b ) }(x,y,z) = (n_{1}+1 ) ( n+n_{2}+n_{3}+e+2 ) P_{n_{1},n_{2},n_{3}}^{(\alpha,\beta,\gamma,\delta,a,b) }(x,y,z), \\& \mathcal{N}_{2,0}^{+}\mathcal{N}_{2,0}P_{n_{1},n_{2},n_{3}}^{ (\alpha+1,\beta,\gamma,\delta,a,b-1 ) }(x,y,z) = (n+n_{2}+n_{3}+e+3 ) ( n+n_{2}+n_{3}+e- \alpha+2 ) \\& \phantom{\mathcal{N}_{2,0}^{+}\mathcal{N}_{2,0}P_{n_{1},n_{2},n_{3}}^{ (\alpha+1,\beta,\gamma,\delta,a,b-1 ) }(x,y,z) =}{}\times P_{n_{1},n_{2},n_{3}}^{ ( \alpha+1,\beta,\gamma,\delta ,a,b-1 )}(x,y,z), \\& \mathcal{N}_{2,0}\mathcal{N}_{2,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha+1,\beta,\gamma,\delta,a,b ) }(x,y,z) = (n+n_{2}+n_{3}+e+3 ) ( n+n_{2}+n_{3}+e- \alpha+2 ) \\& \phantom{\mathcal{N}_{2,0}\mathcal{N}_{2,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha+1,\beta,\gamma,\delta,a,b ) }(x,y,z) =}{}\times P_{n_{1},n_{2},n_{3}}^{ ( \alpha+1,\beta,\gamma,\delta ,a,b )}(x,y,z), \\& \mathcal{N}_{3,0}^{+}\mathcal{N}_{3,0}P_{n_{1},n_{2},n_{3}}^{ (\alpha-1,\beta,\gamma,\delta,a+1,b ) }(x,y,z) = (n_{1}+\alpha ) ( n+n_{2}+n_{3}+e+3 ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha-1,\beta,\gamma,\delta ,a+1,b ) }(x,y,z), \\& \mathcal{N}_{3,0}\mathcal{N}_{3,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma,\delta,a+1,b ) }(x,y,z) = (n_{1}+\alpha ) ( n+n_{2}+n_{3}+e+3 ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta,\gamma,\delta ,a+1,b ) }(x,y,z), \\& \mathcal{N}_{4,0}^{+}\mathcal{N}_{4,0}P_{n_{1}-1,n_{2},n_{3}}^{ (\alpha+1,\beta,\gamma,\delta,a,b ) }(x,y,z) =n_{1} ( n_{1}+\alpha+1 ) P_{n_{1}-1,n_{2},n_{3}}^{ (\alpha+1,\beta ,\gamma,\delta,a,b ) }(x,y,z) , \\& \mathcal{N}_{4,0}\mathcal{N}_{4,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha+1,\beta,\gamma,\delta,a,b-1 ) }(x,y,z) =n_{1} ( n_{1}+\alpha+1 ) P_{n_{1},n_{2},n_{3}}^{ (\alpha+1,\beta ,\gamma,\delta,a,b-1 ) }(x,y,z), \\& \mathcal{N}_{5,0}^{+}\mathcal{N}_{5,0}P_{n_{1}-1,n_{2},n_{3}}^{ (\alpha,\beta,\gamma,\delta,a+1,b ) }(x,y,z) =n_{1} ( n+n_{2}+n_{3}+e-\alpha+3 ) P_{n_{1}-1,n_{2},n_{3}}^{ ( \alpha,\beta,\gamma,\delta,a+1,b ) }(x,y,z), \\& \mathcal{N}_{5,0}\mathcal{N}_{5,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha-1,\beta,\gamma,\delta,a+1,b ) }(x,y,z) =n_{1} ( n+n_{2}+n_{3}+e-\alpha+3 ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha-1,\beta,\gamma,\delta,a+1,b ) }(x,y,z), \\& \mathcal{N}_{6,0}^{+}\mathcal{N}_{6,0}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma,\delta,a,b-1 ) }(x,y,z) = (n_{1}+\alpha ) ( n+n_{2}+n_{3}+e- \alpha+2 ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta,\gamma,\delta ,a,b-1 ) }(x,y,z), \\& \mathcal{N}_{6,0}\mathcal{N}_{6,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha-1,\beta,\gamma,\delta,a,b ) }(x,y,z) = (n_{1}+\alpha ) ( n+n_{2}+n_{3}+e- \alpha+2 ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha-1,\beta,\gamma,\delta ,a,b ) }(x,y,z), \end{aligned}$$
and
$$\begin{aligned}& \mathcal{O}_{1,0}^{+}\mathcal{O}_{1,0}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma-1,\delta-1,a,b ) }(x,y,z) =n_{3} ( n_{3}+\gamma+\delta-1 ) P_{n_{1},n_{2},n_{3}}^{ (\alpha ,\beta,\gamma-1,\delta-1,a,b ) }(x,y,z), \\& \mathcal{O}_{1,0}\mathcal{O}_{1,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma,\delta,a,b ) }(x,y,z) = (n_{3}+1 ) ( n_{3}+\gamma+\delta ) P_{n_{1},n_{2},n_{3}}^{(\alpha,\beta,\gamma,\delta,a,b) }(x,y,z), \\& \mathcal{O}_{2,0}^{+}\mathcal{O}_{2,0}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma+1,\delta-1,a,b ) }(x,y,z) = (n_{3}+\delta ) ( n_{3}+\gamma+\delta+1 ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta,\gamma+1,\delta -1,a,b ) }(x,y,z), \\& \mathcal{O}_{2,0}\mathcal{O}_{2,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma+1,\delta,a,b ) }(x,y,z) = (n_{3}+\delta ) ( n_{3}+\gamma+\delta+1 ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta,\gamma+1,\delta ,a,b ) }(x,y,z), \\& \mathcal{O}_{3,0}^{+}\mathcal{O}_{3,0}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma-1,\delta+1,a,b ) }(x,y,z) = (n_{3}+\gamma ) ( n_{3}+\gamma+\delta+1 ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta,\gamma-1,\delta +1,a,b ) }(x,y,z), \\& \mathcal{O}_{3,0}\mathcal{O}_{3,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma,\delta+1,a,b ) }(x,y,z) = (n_{3}+\gamma ) ( n_{3}+\gamma+\delta+1 ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta,\gamma,\delta +1,a,b ) }(x,y,z), \\& \mathcal{O}_{4,0}^{+}\mathcal{O}_{4,0}P_{n_{1},n_{2},n_{3}-1}^{ (\alpha,\beta,\gamma+1,\delta,a,b ) }(x,y,z) =n_{3} ( n_{3}+\gamma+1 ) P_{n_{1},n_{2},n_{3}-1}^{ (\alpha,\beta ,\gamma+1,\delta,a,b ) }(x,y,z), \\& \mathcal{O}_{4,0}\mathcal{O}_{4,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma+1,\delta-1,a,b ) }(x,y,z) =n_{3} ( n_{3}+\gamma+1 ) P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta ,\gamma+1,\delta-1,a,b ) }(x,y,z), \\& \mathcal{O}_{5,0}^{+}\mathcal{O}_{5,0}P_{n_{1},n_{2},n_{3}-1}^{ (\alpha,\beta,\gamma,\delta+1,a,b ) }(x,y,z) =n_{3} ( n_{3}+\delta+1 ) P_{n_{1},n_{2},n_{3}-1}^{ (\alpha,\beta ,\gamma,\delta+1,a,b ) }(x,y,z), \\& \mathcal{O}_{5,0}\mathcal{O}_{5,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma-1,\delta+1,a,b ) }(x,y,z) =n_{3} ( n_{3}+\delta+1 ) P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta ,\gamma-1,\delta+1,a,b ) }(x,y,z), \\& \mathcal{O}_{6,0}^{+}\mathcal{O}_{6,0}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma,\delta-1,a,b ) }(x,y,z) = (n_{3}+\gamma ) ( n_{3}+\delta ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta,\gamma,\delta -1,a,b ) }(x,y,z), \\& \mathcal{O}_{6,0}\mathcal{O}_{6,0}^{+}P_{n_{1},n_{2},n_{3}}^{ (\alpha,\beta,\gamma-1,\delta,a,b ) }(x,y,z) = (n_{3}+\gamma ) ( n_{3}+\delta ) P_{n_{1},n_{2},n_{3}}^{ ( \alpha,\beta,\gamma-1,\delta ,a,b ) }(x,y,z). \end{aligned}$$