| Laguerre |
\(L_{ls}\) |
\((0,\infty)\) |
\(x^s e^{-x}\) |
\(\frac{1}{l!} x^{-s}e^x (x^{s+l}e^{-x})^{(l)}\) |
\((x^{s+1}e^{-x}L'_{ls})' = - lx^{s+1}e^{-x} L_{ls}\) |
\(\frac{(l+s)!}{l!}\) |
\(\frac{(-1)^l}{l!}\) |
\(-l(l\!+\!s)\) |
\(\exp\big{(}-\frac{xt}{1-t}\big{)}/(1\!-\!t)^{s+1}\) |
| Hermite |
\(H_l\) |
\((-\infty,\infty)\) |
\(e^{-x^2}\) |
\((-1)^le^{x^2} (e^{-x^2})^{(l)}\) |
\((e^{-x^2}H'_l)' = - 2le^{-x^2}H_l\) |
\(2^l l! \sqrt{\pi}\) |
\(2^l\) |
\(0\) |
\(\exp \,(2xt-t^2) \,\,\qquad [/l!]\) |
| Čebyšev I |
\(T_l\) |
\((-1,1)\) |
\(\frac{1}{\sqrt{1-x^2}}\) |
\(\cos(l\arccos{x})\) |
\((\sqrt{1-x^2}\,T'_l)' = - l^2 T_l/\sqrt{1-x^2}\) |
\(\begin{cases} \pi, & l=0\\ \pi/2, & l>0 \end{cases}\) |
\(\begin{cases} 1, & l=0\\ 2^{l-1}, & l>0 \end{cases}\) |
\(0\) |
\(\frac{1-xt}{1-2xt+t^2}\) |
| Čebyšev II |
\(U_l\) |
\((-1,1)\) |
\(\sqrt{1-x^2}\) |
\(\frac{\sin((l+1)\arccos{x})}{\sqrt{1-x^2}}\) |
\(\left[(1-x^2)^{3/2} U'_l\right]' = - l(l\!+\!2)\sqrt{1-x^2} U_l\) |
\(\pi/2\) |
\(2^l\) |
\(0\) |
\(\frac{1}{1-2xt+t^2}\) |
| Legendre |
\(P_l\) |
\((-1,1)\) |
\(1\) |
\(\frac{(-1)^l}{2^l l!}\left[(1-x^2)^l\right]^{(l)}\) |
\(\left[(1-x^2)P'_l\right]' = -l(l+1)P_l\) |
\(\frac{2}{2l+1}\) |
\(\frac{(2l)!}{2^l l!^2}\) |
\(0\) |
\(\frac{1}{\sqrt{1-2xt+t^2}}\) |
| Asociované Legendre |
\(P_{lm}\) |
\((-1,1)\) |
\(1\) |
\(\sqrt{1-x^2}^m P_l^{(m)}\) |
\(\left[(1-x^2)P'_{lm}\right]' = \big{(}\frac{m^2}{1-x^2}- l(l+1)\big{)}P_{lm}\) |
\(\frac{(l+m)!}{(l-m)!}\frac{2}{2l+1}\) |
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