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\form#0:$ L_n^m(x ) = \sum_s $
\form#1:\[ L_n^m(x ) = \sum_s \]
\form#2:\[ L_n^{|m|}(x) L_k(x) x^{|m|} =\sum_{s=0}^{n+|m| +k} d^{|m|}_{nks} L_s(x),\]
\form#3:\[d^{|m|}_{nks} = \int_0^\infty d x e^{-x} L_n^{|m|}(x) L_s(x) L_k(x) x^{|m|}. \]
\form#4:$ d^{|m|}_{nks} $
\form#5:\[ d_{nks}^{|m|} =\sum_{n_1=0}^{n} \sum_{k_1=0}^k \sum_{s_1=0}^s L_{kk_1}^{-1/2} L_{nn_1}^{|m|-1/2} L_{ss_1}^{-1/2} (n_1 + k_1 + s_1 + |m|)!, \]
\form#6:$ $
\form#7:$ \overline{d}^{|m|}_{nkf} $
\form#8:\[ \overline{d}^{|m|}_{nkf} = \sum_{n_1 = 0}^n \sum_{k_1 =0}^k \sum_{f_1=0}^f L_{kk_1}^{-1/2} L_{nn_1}^{|m|-1/2} L_{f f_1}^{-1/2} (n_1 + k_1 + f_1)!. \]
\form#9:$ L_{jl}^p $
\form#10:\[ L_{jl}^{p} = \frac{(-1)^l (p + j + 1/2)!}{(j - l)!(l +p + 1/2 )! l!}. \]
\form#11:\[ L^p+1/2_j(x) = \sum_{l=0} L_{jl}^p L_l(x) \]
\form#12:\[ L^^{p+1/2}_j(x) = \sum_{l=0}^p L_{jl}^p L_l(x) \]
\form#13:\[ L^{p+1/2}_j(x) = \sum_{l=0}^p L_{jl}^p L_l(x) \]
\form#14:\[ \mathcal{A}_{an}^m = \frac{ \sigma_m n!}{(n+|m|)!}\left(\frac{b_a}{2} \right)^{|m|} \kernel{n}(b_a) \]
\form#15:\[ \mathcal{A}_{an}^m = \frac{ \sigma_m n!}{(n+|m|)!}\left(\frac{b_a}{2} \right)^{|m|} \mathcal{k}_{n}(b_a) \]
\form#16:\[ \mathcal{A}_{an}^m = \frac{ \sigma_m n!}{(n+|m|)!}\left(\frac{b_a}{2} \right)^{|m|} \mathcal{K}_{n}(b_a) \]
\form#17:\[ \mathcal{K}_n(x) = \frac{1}{n!} \left(\frac{x}{2} \right)^{2n } e^{- x^2/4} \]
\form#18:\[ e_{ab}^{k}= \int_0^\infty d s_a s_a^{2k} \erf'(\sb) e^{-s_a^2} =e_{ab}^k= \frac{(k-1/2)!}{(-1/2)!(1+\chi^2)^{k+1/2}}. \]
\form#19:\[ e_{ab}^{k}= \int_0^\infty d s_a s_a^{2k} \erf'(s_sb) e^{-s_a^2} =e_{ab}^k= \frac{(k-1/2)!}{(-1/2)!(1+\chi^2)^{k+1/2}}. \]
\form#20:\[ e_{ab}^{k}= \int_0^\infty d s_a s_a^{2k} \erf'(s_b) e^{-s_a^2} =e_{ab}^k= \frac{(k-1/2)!}{(-1/2)!(1+\chi^2)^{k+1/2}}. \]
\form#21:\[ e_{ab}^{k}= \int_0^\infty d s_a s_a^{2k} erf'(s_b) e^{-s_a^2} =e_{ab}^k= \frac{(k-1/2)!}{(-1/2)!(1+\chi^2)^{k+1/2}}. \]
\form#22:\[ e_{ab}^{k}= \int_0^\infty d s_a s_a^{2k} erf'(s_b) e^{-s_a^2} =\frac{(k-1/2)!}{(-1/2)!(1+\chi^2)^{k+1/2}}. \]
\form#23:\[ E_{ab}^{k}= \int_0^\infty d s_a s_a^{2k+1} \erf(s_b)e^{-s_a^2}= \frac{\chi}{2} \sum_{j=0}^k \frac{k!}{j!}e_{ab}^j \]
\form#24:\[ E_{ab}^{k}= \int_0^\infty d s_a s_a^{2k+1} erf(s_b)e^{-s_a^2}= \frac{\chi}{2} \sum_{j=0}^k \frac{k!}{j!}e_{ab}^j \]
\form#25:$ \alpha_{ab1}^{pl}$
\form#26:\[ \alpha_{ab1}^{pl} =4 \frac{\sigma}{\tau}l(\tau -2)-p (p -2 \sigma +4\frac{ \sigma}{\tau} +1) \]
\form#27:$ \alpha_{ab2}^{pl}$
\form#28:\[ \alpha_{ab2}^{pl} =\frac{\sigma}{\tau} \left(4 l^2+4 l (p-1)+\frac{3}{2} p(p-1) \]
\form#29:\[ \alpha_{ab2}^{pl} =\frac{\sigma}{\tau} \left(4 l^2+4 l (p-1)+\frac{3}{2} p(p-1) \right)\]
\form#30:$ \alpha_{ab1}^{'pl}$
\form#31:$ \alpha_{ab2}^{'pl}$
\form#32:\[ a^{'pl}_{ab2}= - \frac{\sigma}{\tau} \left(4l^2+4 l (p-1)+\frac{3}{2}p (p-1) \right) \]
\form#33:\[ \]
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