Fórmula general
$$ H_n(x)=(-1)^n e^{x^2 / 2} \frac{d^n}{d x^n} e^{-x^2 / 2} $$
Primers valors
$$ \begin{aligned} & H_0(x)=1 \\ & H_1(x)=2 x \\ & H_2(x)=4 x^2-2 \\ & H_3(x)=8 x^3-12 x \\ & H_4(x)=16 x^4-48 x^2+12 \\ & H_5(x)=32 x^5-160 x^3+120 x \end{aligned} $$
Ortogonalitat
$$ \int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2} d x=n!2^n \sqrt{\pi} \delta_{n m} $$
Fórmula general
$$ P_l(x)=\frac{(-1)^l}{2^l l!} \frac{d^l}{d x^l}\left(1-x^2\right)^l $$
Primers valors
$$ \begin{array}{ll} P_0(x)=1 & P_3(x)=\frac{1}{2}\left(5 x^3-3 x\right) \\ P_1(x)=x & P_4(x)=\frac{1}{8}\left(35 x^4-30 x^2+3\right) \\ P_2(x)=\frac{1}{2}\left(3 x^2-1\right) & P_5(x)=\frac{1}{8}\left(63 x^5-70 x^3+15 x\right) \end{array} $$
Fórmula general
$$ P_{l, m}(x)=(-1)^m \sqrt{\left(1-x^2\right)^m} \frac{d^m}{d x^m} P_l(x) $$
Primers valors
$$ \begin{array}{ll} P_{0,0}(x)=1 & P_{2,0}(x)=\frac{1}{2}\left(3 x^2-1\right) \\ P_{1,1}(x)=-\sqrt{1-x^2} & P_{3,3}(x)=-15\left(\sqrt{1-x^2}\right)^3 \\ P_{1,0}(x)=x & P_{3,2}(x)=15 x\left(1-x^2\right) \\ P_{2,2}(x)=3\left(1-x^2\right) & P_{3,1}(x)=-\frac{3}{2}\left(5 x^2-1\right) \sqrt{1-x^2} \\ P_{2,1}(x)=-3 x \sqrt{1-x^2} & P_{3,0}(x)=\frac{1}{2}\left(5 x^3-3 x\right) \end{array} $$
Fórmula general