Editing Rodrigues' formula (section)

== Examples ==
{| class="wikitable"
|+
!Family
!<math>[a,b]</math>
!<math>w</math>
!<math>W</math>
!<math>A</math>
!<math>B</math>
!<math>c_n</math>
|-
|[[Legendre polynomials|Legendre]] <math>P_n</math>
|<math>[-1,+1]</math>
|<math>1</math>
|<math>1-x^2</math>
|<math>-2x</math>
|<math>1-x^2</math>
|<math>\frac{(-1)^n}{2^n n!}</math>
|-
|[[Chebyshev polynomials|Chebyshev]] (of the first kind) <math>T_n</math>
|<math>[-1,+1]</math>
|<math>1/\sqrt{1-x^2}</math>
|<math>\sqrt{1-x^2}</math>
|<math>-x</math>
|<math>1-x^2</math>
|<math>\frac{(-1)^n}{(2n-1)!!}</math>
|-
|[[Chebyshev polynomials|Chebyshev]] (of the second kind) <math>U_n</math>
|<math>[-1,+1]</math>
|<math>\sqrt{1-x^2}</math>
|<math>(1-x^2)^{3/2}</math>
|<math>-3x</math>
|<math>1-x^2</math>
|<math>\frac{(-1)^n (n+1)}{(2n+1)!!} </math>
|-
|[[Gegenbauer polynomials|Gegenbauer/ultraspherical]] <math>C_n^{(\alpha)}(x)</math>
|<math>[-1,+1]</math>
|<math>(1-x)^{\alpha-1/2} (1+x)^{\alpha-1/2} </math>
|<math>(1-x)^{\alpha+1/2} (1+x)^{\alpha+1/2} </math>
|<math>-(2\alpha + 1)x </math>
|<math>1-x^2</math>
|<math>\frac{(-1)^n (2\alpha)_n}{(\alpha+\frac{1}{2})_{n} 2^nn!}</math>
|-
|[[Jacobi polynomials|Jacobi]] <math>P_n^{(\alpha, \beta)}</math>
|<math>[-1,+1]</math>
|<math>(1-x)^\alpha (1+x)^\beta</math>
|<math>(1-x)^{\alpha+1} (1+x)^{\beta +1}</math>
|<math>( \beta - \alpha ) - (\alpha+ \beta + 2) x</math>
|<math>1-x^2</math>
|<math>\frac{(-1)^n}{2^nn!}</math>
|-
|associated [[Laguerre polynomials|Laguerre]] <math>L^{(\alpha)}_n</math>
|<math>[0, \infty)</math>
|<math>x^\alpha e^{-x}</math>
|<math>x^{\alpha+1} e^{-x}</math>
|<math>\alpha + 1 - x</math>
|<math>x</math>
|<math>\frac{1}{n!}</math>
|-
|[[Hermite polynomials|physicist's Hermite]] <math>H_n</math>
|<math>(-\infty, +\infty)</math>
|<math>e^{-x^2}</math>
|<math>e^{-x^2}</math>
|<math>-2x</math>
|<math>1</math>
|<math>(-1)^n</math>
|}
Similar formulae hold for many other sequences of [[orthogonal functions]] arising from [[Sturm–Liouville equation]]s, and these are also called the Rodrigues formula (or Rodrigues' type formula), especially when the resulting sequence is polynomial.

=== Legendre ===
Rodrigues stated his formula for Legendre polynomials <math>P_n</math>:
<math display="block">P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} \!\left[ (x^2 -1)^n \right]\!.</math><math display="block">(1 - x^2) P_n''(x) - 2 x P_n'(x) + n (n + 1) P_n(x) = 0</math>For Legendre polynomials, the generating function is defined as <math display="”block”">G(x,u)=\sum_{n=0}^\infty u^nP_n(x)</math>.

The contour integral gives the '''Schläfli integral'''<ref>{{Citation |last=Schläfli |first=Ludwig |title=Über die zwei Heineschen Kugelfunktionen mit beliebigem Parameter und ihre ausnahmslose Darstellung durch bestimmte Integrale |date=1881 |work=Gesammelte Mathematische Abhandlungen |pages=317–392 |url=https://doi.org/10.1007/978-3-0348-4116-0_27 |place=Basel |publisher=Springer Basel |isbn=978-3-0348-4044-6}}</ref> for Legendre polynomials:<math display="block">P_n(x) = \frac{1}{2\pi i 2^n} \oint_C \frac{(t^2-1)^n}{(t-x)^{n+1}} dt</math> Summing up the integrand,<math display="block">G(x,u) = \frac{1}{\sqrt{1 - 2ux + u^2}} \frac{1}{2\pi i} \oint_C \left(\frac{1}{t - t_-} - \frac{1}{t - t_+}\right) dt</math>where <math>t_\pm = \frac{1}{u} (1 \pm \sqrt{1 - 2ux + u^2})</math>. For small <math>u</math>, we have <math>t_- \approx x, t_+ \to \infty</math>, which heuristically suggests that the integral should be the residue around <math>t_-</math>, thus giving<math display="block">G(x,u) = \frac{1}{\sqrt{1 - 2ux + u^2}}</math>

=== Hermite ===
Physicist's [[Hermite polynomials]]:<math display="block">H_n(x)=(-1)^n e^{x^2} \frac{d^n}{dx^n} \!\left[e^{-x^2}\right] = \left(2x-\frac{d}{dx} \right)^n\cdot 1.</math><math display="block">H_n'' - 2xH_n' + 2nH_n = 0</math>

The generating function is defined as<math display="block">G(x,u)=\sum_{n=0}^\infty \frac{H_n(x)}{n!}\, u^n.</math>The contour integral gives<math display="block"> H_n(x)=(-1)^n e^{x^2}\frac{n!}{2\pi i}\oint_C \frac{e^{-t^2}}{(t-x)^{n+1}}\,dt. </math><math display="block"> \begin{aligned} G(x,u) 
&= \sum_{n=0}^\infty \frac{(-1)^n e^{x^2}}{n!}\frac{n!}{2\pi i}\, u^n \oint_C \frac{e^{-t^2}}{(t-x)^{n+1}}\,dt \\ 
&= e^{x^2}\frac{1}{2\pi i}\oint_C e^{-t^2}\left(\sum_{n=0}^\infty \frac{(-1)^n u^n}{(t-x)^{n+1}}\right)dt \\
&= e^{x^2}\frac{1}{2\pi i}\oint_C e^{-t^2} \frac{1}{t-x+u}\\
&= e^{x^2}\, e^{-(x-u)^2} \\
& = e^{2xu- u^2}
\end{aligned} </math>

=== Laguerre ===
For associated [[Laguerre polynomials]],<math display="block">L_n^{(\alpha)}(x) = {x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right)
= \frac{x^{-\alpha}}{n!}\left( \frac{d}{dx}-1\right)^nx^{n+\alpha}.</math><math display="block">xL^{(\alpha)}_n(x)'' + (\alpha + 1 - x)L^{(\alpha)}_n(x)' + nL^{(\alpha)}_n(x) = 0~.</math>

The generating function is defined as<math display="block">G(x,u) := \sum_{n=0}^\infty  u^n L^{(\alpha)}_n(x)</math>By the same method, we have <math>G(x,u) = \frac{1}{(1-u)^{\alpha+1}} e^{-\frac{ux}{1-u}}</math>.

=== Jacobi ===
<math display="block"> P_n^{(\alpha,\beta)}(x) = \frac{(-1)^n}{2^n n!} (1-x)^{-\alpha} (1+x)^{-\beta} \frac{d^n}{dx^n} \left\{ (1-x)^\alpha (1+x)^\beta \left (1 - x^2 \right )^n \right\}.</math><math display="block"> \left (1-x^2 \right)P_n^{(\alpha,\beta)}{}'' + ( \beta-\alpha - (\alpha + \beta + 2)x )P_n^{(\alpha,\beta)}{}' + n(n+\alpha+\beta+1) P_n^{(\alpha,\beta)} = 0.</math>

: <math> \sum_{n=0}^\infty P_n^{(\alpha,\beta)}(x) u^n = 2^{\alpha + \beta} R^{-1} (1 - u + R)^{-\alpha} (1 + u + R)^{-\beta}, </math>

where <math display="inline"> R = \sqrt{1 - 2ux + u^2}  </math>, and the [[Principal branch|branch]] of square root is chosen so that <math>R(x, 0) = 1</math>.