Editing Classical orthogonal polynomials (section)

===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).