∫ a b p m ( x ) p n ( x ) w ( x ) 𝑑 x = 0 , m ≠ n formulae-sequence superscript subscript 𝑎 𝑏 subscript 𝑝 𝑚 𝑥 subscript 𝑝 𝑛 𝑥 𝑤 𝑥 differential-d 𝑥 0 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}p_{m}(x)p_{n}(x)w(x)\,dx% =0,\quad m\neq n}}} {\displaystyle \int_a^bp_m(x)p_n(x)w(x)\,dx=0,\quad m\neq n }
0 < ∫ a b x 2 n w ( x ) 𝑑 x < ∞ for all formulae-sequence 0 superscript subscript 𝑎 𝑏 superscript 𝑥 2 𝑛 𝑤 𝑥 differential-d 𝑥 for all {\displaystyle{\displaystyle{\displaystyle 0<\int_{a}^{b}x^{2n}w(x)\,dx<\infty% \quad\textrm{for all}}}} {\displaystyle 0<\int_a^bx^{2n}w(x)\,dx<\infty\quad\textrm{for all} }
0 < ∫ a b x 2 n 𝑑 α ( x ) < ∞ for all formulae-sequence 0 superscript subscript 𝑎 𝑏 superscript 𝑥 2 𝑛 differential-d 𝛼 𝑥 for all {\displaystyle{\displaystyle{\displaystyle 0<\int_{a}^{b}x^{2n}\,d\alpha(x)<% \infty\quad\textrm{for all}}}} {\displaystyle 0<\int_a^bx^{2n}\,d\alpha(x)<\infty\quad\textrm{for all} }
∑ x ∈ X p m ( x ) p n ( x ) w x = 0 , m ≠ n formulae-sequence subscript 𝑥 𝑋 subscript 𝑝 𝑚 𝑥 subscript 𝑝 𝑛 𝑥 subscript 𝑤 𝑥 0 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x\in X}p_{m}(x)p_{n}(x)w_{x}=0% ,\quad m\neq n}}} {\displaystyle \sum_{x\in X}p_m(x)p_n(x)w_x=0,\quad m\neq n }
σ n = ∫ a b { p n ( x ) } 2 w ( x ) 𝑑 x subscript 𝜎 𝑛 superscript subscript 𝑎 𝑏 superscript subscript 𝑝 𝑛 𝑥 2 𝑤 𝑥 differential-d 𝑥 {\displaystyle{\displaystyle{\displaystyle\sigma_{n}=\int_{a}^{b}\left\{p_{n}(% x)\right\}^{2}w(x)\,dx}}} {\displaystyle \sigma_n=\int_a^b\left\{p_n(x)\right\}^2w(x)\,dx }
σ n = ∑ x ∈ X { p n ( x ) } 2 w x subscript 𝜎 𝑛 subscript 𝑥 𝑋 superscript subscript 𝑝 𝑛 𝑥 2 subscript 𝑤 𝑥 {\displaystyle{\displaystyle{\displaystyle\sigma_{n}=\sum_{x\in X}\left\{p_{n}% (x)\right\}^{2}w_{x}}}} {\displaystyle \sigma_n=\sum_{x\in X}\left\{p_n(x)\right\}^2w_x }
p n ( x ) = k n x n + lower order terms subscript 𝑝 𝑛 𝑥 subscript 𝑘 𝑛 superscript 𝑥 𝑛 lower order terms {\displaystyle{\displaystyle{\displaystyle p_{n}(x)=k_{n}x^{n}+\,\textrm{lower% order terms}}}} {\displaystyle p_n(x)=k_nx^n+\,\textrm{lower order terms} }
δ m , n := { 0 , m ≠ n , 1 , m = n , assign Kronecker-delta 𝑚 𝑛 cases 0 𝑚 𝑛 1 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\,\delta_{m,n}:=\left\{\begin{array}% []{ll}0,&m\neq n,\\ 1,&m=n,\end{array}\right.}}} {\displaystyle \,\Kronecker{m}{n}:=\left\{\begin{array}{ll}0, &m\neq n,\[5mm] 1, & m=n,\end{array}\right. }
∫ a b p m ( x ) p n ( x ) w ( x ) 𝑑 x = σ n δ m , n superscript subscript 𝑎 𝑏 subscript 𝑝 𝑚 𝑥 subscript 𝑝 𝑛 𝑥 𝑤 𝑥 differential-d 𝑥 subscript 𝜎 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}p_{m}(x)p_{n}(x)w(x)\,dx% =\sigma_{n}\,\delta_{m,n}}}} {\displaystyle \int_a^bp_m(x)p_n(x)w(x)\,dx=\sigma_n\,\Kronecker{m}{n} }
∑ x ∈ X p m ( x ) p n ( x ) w x = σ n δ m , n subscript 𝑥 𝑋 subscript 𝑝 𝑚 𝑥 subscript 𝑝 𝑛 𝑥 subscript 𝑤 𝑥 subscript 𝜎 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x\in X}p_{m}(x)p_{n}(x)w_{x}=% \sigma_{n}\,\delta_{m,n}}}} {\displaystyle \sum_{x\in X}p_m(x)p_n(x)w_x=\sigma_n\,\Kronecker{m}{n} }