Editing Classical orthogonal polynomials (section)

== Definition ==

In general, the orthogonal polynomials <math>P_n</math> with respect to a weight <math>W:\mathbb R \rightarrow \mathbb R^+ </math> satisfy

:<math>\begin{align}
&\deg P_n = n~, \quad n = 0,1,2,\ldots\\
&\int P_m(x) \, P_n(x) \, W(x)\,dx = 0~, \quad m \neq n~.
\end{align}</math>

The relations above define <math>P_n</math> up to multiplication by a number. Various normalisations are used to fix the constant, e.g.

:<math> \int P_n(x)^2 W(x)\,dx = 1~.</math>

The classical orthogonal polynomials correspond to the following three families of weights:

:<math>\begin{align}
\text{(Jacobi)}\quad &W(x) = \begin{cases} 
  (1 - x)^\alpha (1+x)^\beta~, & -1 \leq x \leq 1 \\
  0~, &\text{otherwise}
\end{cases}  \\
\text{(Hermite)}\quad  &W(x) = \exp(- x^2) \\
\text{(Laguerre)}\quad &W(x) = \begin{cases}
  x^\alpha \exp(- x)~, & x \geq 0 \\
  0~, & \text{otherwise}
\end{cases}      
\end{align}</math>

The standard normalisation (also called ''standardization'') is detailed below.

===Jacobi polynomials===
{{main article|Jacobi polynomials}}

For <math>\alpha,\,\beta>-1</math> the Jacobi polynomials are given by the formula

:<math>P_n^{(\alpha,\beta)} (z)
= \frac{(-1)^n}{2^n n!} (1-z)^{-\alpha} (1+z)^{-\beta}
\frac{d^n}{dz^n} \left\{ (1-z)^\alpha (1+z)^\beta (1 - z^2)^n \right\}~. </math>

They are normalised (standardized) by

:<math>P_n^{(\alpha, \beta)} (1) = {n+\alpha\choose n},</math>

and satisfy the orthogonality condition

:<math>\begin{align}
&\int_{-1}^1 (1-x)^{\alpha} (1+x)^{\beta} 
P_m^{(\alpha,\beta)} (x)P_n^{(\alpha,\beta)} (x) \; dx \\
= {} &
\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}
\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!} \delta_{nm}.
\end{align}
</math>

The Jacobi polynomials are solutions to the differential equation

:<math>
(1-x^2)y'' + ( \beta-\alpha - (\alpha + \beta + 2)x )y'+ n(n+\alpha+\beta+1) y = 0~.
</math>

==== Important special cases ====

The Jacobi polynomials with <math>\alpha=\beta</math> are called the [[Gegenbauer polynomials]] (with parameter <math>\gamma = \alpha+1/2</math>)

For <math>\alpha=\beta=0</math>, these are called the [[Legendre polynomials]] (for which the interval of orthogonality is [&minus;1,&nbsp;1] and the weight function is simply 1):

:<math>
P_0(x) = 1,\, P_1(x) = x,\,P_2(x) = \frac{3x^2-1}{2},\,
P_3(x) = \frac{5x^3-3x}{2},\ldots</math>

For <math>\alpha=\beta=\pm 1/2</math>, one obtains the [[Chebyshev polynomials]] (of the second and first kind, respectively).

===Hermite polynomials===
{{main article|Hermite polynomials}}

The Hermite polynomials are defined by<ref>other conventions are also used; see [[Hermite polynomials]].</ref>

:<math> H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}=e^{x^2/2}\bigg (x-\frac{d}{dx} \bigg )^n e^{-x^2/2}</math>

They satisfy the orthogonality condition

:<math> \int_{-\infty}^\infty H_n(x) H_m(x) e^{-x^2} \, dx = \sqrt{\pi} 2^n n! \delta_{mn}~, </math>

and the differential equation

:<math>y'' - 2xy' + 2n\,y = 0~.</math>

===Laguerre polynomials===
{{main article|Laguerre polynomials}}

The generalised Laguerre polynomials are defined by

: <math>L_n^{(\alpha)}(x)=
{x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right)</math>

(the classical Laguerre polynomials correspond to <math>\alpha=0</math>.)

They satisfy the orthogonality relation

: <math>\int_0^\infty x^\alpha e^{-x} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x) \, dx=\frac{\Gamma(n+\alpha+1)}{n!}\delta_{n,m}~,</math>

and the differential equation

:<math>
x\,y'' + (\alpha +1 - x)\,y' + n\,y = 0~.</math>