Editing Classical orthogonal polynomials (section)

== Table of classical orthogonal polynomials ==

The following table summarises the properties of the classical orthogonal polynomials.<ref>See {{harvtxt|Abramowitz|Stegun|1983}}</ref>

{| border="1" cellspacing="0" cellpadding="5" style="margin:1em auto;"
|-----
! Name, and conventional symbol
! [[Chebyshev polynomials|Chebyshev]], <math>\ T_n</math>
! [[Chebyshev polynomials|Chebyshev]]<br>(second kind), <math>\ U_n</math>
! [[Legendre polynomials|Legendre]], <math>\ P_n</math>
! [[Hermite polynomials|Hermite]], <math>\ H_n</math>
|-----
| Limits of orthogonality<ref>i.e. the edges of the support of the weight ''W''.</ref>
| <math>-1, 1</math>
| <math>-1, 1</math>
| <math>-1, 1</math> 
| <math>-\infty, \infty</math>
|-----
| Weight, <math>W(x)</math> 
| <math>(1-x^2)^{-1/2}</math>
| <math>(1-x^2)^{1/2}</math>
| <math>1</math> 
| <math>e^{-x^2}</math>
|-----
| Standardization 
| <math>T_n(1)=1</math>
| <math>U_n(1)=n+1</math>
| <math>P_n(1)=1</math> 
| Lead term <math>=2^n</math>
|-----
| Square of norm <ref><math>h_n = \int P_n^2(x) W(x) \, dx</math></ref>
| <math>\left\{
\begin{matrix}
\pi   &:~n=0 \\
\pi/2 &:~n\ne 0
\end{matrix}\right.
</math>
| <math>\pi/2</math> 
| <math>\frac{2}{2n+1}</math>
| <math>2^n\,n!\,\sqrt{\pi}</math>
|-----
| Leading term <ref>The leading coefficient ''k''<sub>''n''</sub> of <math> P_n(x) = k_n x^n + k'_n x^{n-1} + \cdots + k^{(n)} </math></ref>
| <math>2^{n-1}</math>
| <math>2^n</math>
| <math>\frac{(2n)!}{2^n\,(n!)^2}</math>
| <math>2^n</math>
|-----
| Second term, <math>k'_n</math> 
| <math>0</math>
| <math>0</math>
| <math>0</math> 
| <math>0</math>
|-----
| <math>Q</math> 
| <math>1-x^2</math>
| <math>1-x^2</math>
| <math>1-x^2</math> 
| <math>1</math>
|-----
| <math>L</math> 
| <math>-x</math>
| <math>-3x</math>
| <math>-2x</math> 
| <math>-2x</math>
|-----
| <math>R(x) =e^{\int \frac{L(x)}{Q(x)}\,dx}</math>
| <math>(1-x^2)^{1/2}</math> 
| <math>(1-x^2)^{3/2}</math>
| <math>1-x^2</math> 
| <math>e^{-x^2}</math>
|-----
| Constant in diff. equation, <math>\lambda_n</math>
| <math>n^2</math> 
| <math>n(n+2)</math>
| <math>n(n+1)</math> 
| <math>2n</math>
|-----
| Constant in Rodrigues' formula, <math>e_n</math>
| <math>(-2)^n\,\frac{\Gamma(n+1/2)}{\sqrt{\pi}}</math>
| <math>2(-2)^n\,\frac{\Gamma(n+3/2)}{(n+1)\,\sqrt{\pi}}</math>
| <math>(-2)^n\,n!</math> 
| <math>(-1)^n</math>
|-----
| Recurrence relation, <math>a_n</math>
| <math>2</math> 
| <math>2</math>
| <math>\frac{2n+1}{n+1}</math> 
| <math>2</math>
|-----
| Recurrence relation, <math>b_n</math>
| <math>0</math> 
| <math>0</math>
| <math>0</math> 
| <math>0</math>
|-----
| Recurrence relation, <math>c_n</math>
| <math>1</math> 
| <math>1</math>
| <math>\frac{n}{n+1}</math> 
| <math>2n</math>
|}

{| border="1" cellspacing="0" cellpadding="5" style="margin:1em auto;"
|-----
! Name, and conventional symbol
! [[Laguerre polynomials|Associated Laguerre]], <math>L_n^{(\alpha)}</math>
! [[Laguerre polynomials|Laguerre]], <math>\ L_n</math>
|-----
| Limits of orthogonality 
| <math>0, \infty</math>
| <math>0, \infty</math>
|-----
| Weight, <math>W(x)</math> 
| <math>x^{\alpha}e^{-x}</math>
| <math>e^{-x}</math>
|-----
| Standardization
| Lead term <math>=\frac{(-1)^n}{n!}</math>
| Lead term <math>=\frac{(-1)^n}{n!}</math>
|-----
| Square of norm, <math>h_n</math>
| <math>\frac{\Gamma(n+\alpha+1)}{n!}</math>
| <math>1</math>
|-----
| Leading term, <math>k_n</math> 
| <math>\frac{(-1)^n}{n!}</math>
| <math>\frac{(-1)^n}{n!}</math>
|-----
| Second term, <math>k'_n</math>
| <math>\frac{(-1)^{n+1}(n+\alpha)}{(n-1)!}</math>
| <math>\frac{(-1)^{n+1}n}{(n-1)!}</math>
|-----
| <math>Q</math> 
| <math>x</math>
| <math>x</math>
|-----
| <math>L</math> 
| <math>\alpha+1-x</math>
| <math>1-x</math>
|-----
| <math>R(x) =e^{\int \frac{L(x)}{Q(x)}\,dx}</math>
| <math>x^{\alpha+1}\,e^{-x}</math> 
| <math>x\,e^{-x}</math>
|-----
| Constant in diff. equation, <math>\lambda_n</math>
| <math>n</math> 
| <math>n</math>
|-----
| Constant in Rodrigues' formula, <math>e_n</math>
| <math>n!</math> 
| <math>n!</math>
|-----
| Recurrence relation, <math>a_n</math>
| <math>\frac{-1}{n+1}</math> 
| <math>\frac{-1}{n+1}</math>
|-----
| Recurrence relation, <math>b_n</math>
| <math>\frac{2n+1+\alpha}{n+1}</math>
| <math>\frac{2n+1}{n+1}</math>
|-----
| Recurrence relation, <math>c_n</math>
| <math>\frac{n+\alpha}{n+1}</math> 
| <math>\frac{n}{n+1}</math>
|}

{| border="1" cellspacing="0" cellpadding="5" style="margin:1em auto;"
|-----
! Name, and conventional symbol
! [[Gegenbauer polynomials|Gegenbauer]], <math>C_n^{(\alpha)}</math>
! [[Jacobi polynomials|Jacobi]], <math>P_n^{(\alpha, \beta)}</math>
|-----
| Limits of orthogonality 
| <math>-1, 1</math>
| <math>-1, 1</math>
|-----
| Weight, <math>W(x)</math> 
| <math>(1-x^2)^{\alpha-1/2}</math>
| <math>(1-x)^\alpha(1+x)^\beta</math>
|-----
| Standardization
| <math>C_n^{(\alpha)}(1)=\frac{\Gamma(n+2\alpha)}{n!\,\Gamma(2\alpha)}</math> if <math>\alpha\ne0</math>
| <math>P_n^{(\alpha, \beta)}(1)=\frac{\Gamma(n+1+\alpha)}{n!\,\Gamma(1+\alpha)}</math>
|-----
| Square of norm, <math>h_n</math>
| <math>\frac{\pi\,2^{1-2\alpha}\Gamma(n+2\alpha)}{n!(n+\alpha)(\Gamma(\alpha))^2}</math>
| <math>\frac{2^{\alpha+\beta+1}\,\Gamma(n\!+\!\alpha\!+\!1)\,\Gamma(n\!+\!\beta\!+\!1)}
{n!(2n\!+\!\alpha\!+\!\beta\!+\!1)\Gamma(n\!+\!\alpha\!+\!\beta\!+\!1)}</math>
|-----
| Leading term, <math>k_n</math>
| <math>\frac{\Gamma(2n+2\alpha)\Gamma(1/2+\alpha)}{n!\,2^n\,\Gamma(2\alpha)\Gamma(n+1/2+\alpha)}</math>
| <math>\frac{\Gamma(2n+1+\alpha+\beta)}{n!\,2^n\,\Gamma(n+1+\alpha+\beta)}</math>
|-----
| Second term, <math>k'_n</math> 
| <math>0</math>
| <math>\frac{(\alpha-\beta)\,\Gamma(2n+\alpha+\beta)}{(n-1)!\,2^n\,\Gamma(n+1+\alpha+\beta)}</math>
|-----
| <math>Q</math> 
| <math>1-x^2</math>
| <math>1-x^2</math>
|-----
| <math>L</math> 
| <math>-(2\alpha+1)\,x</math>
| <math>\beta-\alpha-(\alpha+\beta+2)\,x</math>
|-----
| <math>R(x) =e^{\int \frac{L(x)}{Q(x)}\,dx}</math>
| <math>(1-x^2)^{\alpha+1/2}</math>
| <math>(1-x)^{\alpha+1}(1+x)^{\beta+1}</math>
|-----
| Constant in diff. equation, <math>\lambda_n</math>
| <math>n(n+2\alpha)</math> 
| <math>n(n+1+\alpha+\beta)</math>
|-----
| Constant in Rodrigues' formula, <math>e_n</math>
| <math>\frac{(-2)^n\,n!\,\Gamma(2\alpha)\,\Gamma(n\!+\!1/2\!+\!\alpha)}
{\Gamma(n\!+\!2\alpha)\Gamma(\alpha\!+\!1/2)}</math>
| <math>(-2)^n\,n!</math>
|-----
| Recurrence relation, <math>a_n</math>
| <math>\frac{2(n+\alpha)}{n+1}</math>
| <math>\frac{(2n+1+\alpha+\beta)(2n+2+\alpha+\beta)}{2(n+1)(n+1+\alpha+\beta)}</math>
|-----
| Recurrence relation, <math>b_n</math>
| <math>0</math>
| <math>\frac{({\alpha}^2-{\beta}^2)(2n+1+\alpha+\beta)}{2(n+1)(2n+\alpha+\beta)(n+1+\alpha+\beta)}</math>
|-----
| Recurrence relation, <math>c_n</math>
| <math>\frac{n+2{\alpha}-1}{n+1}</math>
| <math>\frac{(n+\alpha)(n+\beta)(2n+2+\alpha+\beta)}{(n+1)(n+1+\alpha+\beta)(2n+\alpha+\beta)}</math>
|}