Editing Rodrigues' formula

{{Short description|Formula for the Legendre polynomials}}
{{for|the 3-dimensional rotation formula|Rodrigues' rotation formula}}
In [[mathematics]], '''Rodrigues' formula''' (formerly called the '''Ivory–Jacobi formula''') generates the [[Legendre polynomials]]. It was independently introduced by {{harvs|txt|authorlink=Olinde Rodrigues|first=Olinde|last=Rodrigues|year=1816}}, {{harvs|txt|authorlink=James Ivory (mathematician)|first=Sir James|last= Ivory|year=1824}} and {{harvs|txt|authorlink=Carl Gustav Jacob Jacobi|first=Carl Gustav|last=Jacobi|year=1827}}. The name "Rodrigues formula" was introduced by [[Eduard Heine|Heine]] in 1878, after Hermite pointed out in 1865 that Rodrigues was the first to discover it. The term is also used to describe similar formulas for other [[orthogonal polynomials]]. {{harvtxt|Askey|2005}} describes the history of the Rodrigues formula in detail.

==Statement==
Let <math>(P_n(x))_{n=0}^\infty</math> be a [[sequence]] of orthogonal polynomials on the interval <math>[a, b]</math> with respect to [[weight function]] <math>w(x)</math>. That is, they have degrees <math>deg(P_n) = n</math>, satisfy the orthogonality condition
<math display="block">\int_a^b P_m(x) P_n(x) w(x) \, dx = K_n \delta_{m,n}</math>
where <math>K_n</math> are nonzero constants depending on <math>n</math>, and <math>\delta_{m,n}</math> is the [[Kronecker delta]]. The interval <math>[a, b]</math> may be infinite in one or both ends.

{{Math theorem
| name = Rodrigues' type formula
| note = 
| math_statement = If
<math display="block">w(x)=W(x)/B(x), \quad \frac{W'(x)}{W(x)} = \frac{A(x)}{B(x)},</math>
where <math>A(x)</math> is a [[polynomial]] with [[degree of a polynomial|degree]] at most 1 and <math>B(x)</math> is a polynomial with degree at most 2, and
<math display="block">\lim_{x \to a} x^k W(x) = 0, \qquad \lim_{x \to b}  x^k W(x) = 0.</math>
for any <math>k = 0, 1, 2, \dots</math>.

Then, if <math>\frac{d^n}{dx^n} \!\left[ B(x)^n w(x)\right] \neq 0</math> for all <math>n = 0, 1, 2, \dots</math>, then
<math display="block">P_n(x) = \frac{c_n}{w(x)} \frac{d^n}{dx^n} \!\left[ B(x)^n w(x)\right],</math>
for some constants <math>c_n</math>.
}}

{{Math proof|title=Proof|proof= 

Let <math display="inline">F_k := \frac 1w D_x^k(B^n w)</math>, then <math display="inline">F_k = B^{n-k} p_k</math> for all <math display="inline">k \in 0:n</math> for some polynomials <math display="inline">p_k</math>, such that <math display="inline">deg(p_k) \leq k</math>. Proven by induction on <math display="inline">k</math>: <math display="block">
 F_{k+1} = B^{n-k-1}(B p_k' + (n-k)B' p_k + (A-B')p_k)
 </math>

Let <math display="inline">Q_n := \frac 1w D_x^n(B^n w)</math>. We have shown that <math display="inline">Q_n</math> is a polynomial of degree <math>\leq n</math>. With integration by parts, we have for all <math display="inline">n > m</math>, <math display="block">
 \int_a^b Q_m Q_n w dx = \int_a^b B^n w (D_x^n Q_m) dx = 0
 </math> since <math display="inline">D_x^n Q_m=0</math>. Thus, <math display="inline">Q_0, Q_1, \dots</math> make up an orthogonal polynomial series with respect to <math display="inline">w</math>. Thus, <math display="inline">P_n = c_n Q_n</math> for some constants <math display="inline">c_n</math>.
}}

{{Math theorem
| name = Differential equation
| note = 
| math_statement = <math display="block">B(x) \frac{d^2}{dx^2} P_n(x) + A(x) \frac{d}{dx}  P_n(x) + \lambda_n P_n(x) = 0</math>

<math display="block">\lambda_n = -\frac{1}{2}n(n-1)B''-nA'</math>
}}

{{Math proof|title=Proof|proof= 
When <math>n = 0</math>, it is trivial. When <math>n = 1</math>, it simplifies to <math>AP_1' = A'P_1</math>, which is true since <math>P_1 = \frac{c_1}{w}(Bw)' = c_1A</math>. So assume <math>n \geq 2</math>. Define <math>I_n(x) = \frac{d^n}{dx^n}(B^n(x) w(x))</math>, then by direct computation and simplification, the equation to be proven is equivalent to

<math display="block">\frac{d^2}{dx^2} (B(x) I_n(x)) - \frac{d}{dx} (A(x) I_n(x)) + \lambda_n I_n(x) = 0</math>

By Leibniz differentiation rule, we have

<math display="block">B(x) \frac{d^n}{dx^n} y = \frac{d^n}{dx^n} (B(x) y) - n \frac{d^{n-1}}{dx^{n-1}} (B'(x) y) + \frac{n(n-1)}{2} \frac{d^{n-2}}{dx^{n-2}} (B'' y)</math>

<math display="block">A(x) \frac{d^n}{dx^n} y = \frac{d^n}{dx^n} (A(x) y) - n \frac{d^{n-1}}{dx^{n-1}} (A' y)</math>

for arbitrary <math>y</math>. This allows us to move <math>A(x), B(x)</math> to the other side of the <math>n</math>-th derivative. Set <math>y = B^n(x) w(x) </math>, and define

<math display="block">J(x) = \frac{d^2}{dx^2} (B(x) y(x)) - n \frac{d}{dx} (B'(x) y(x)) + \frac{n(n-1)}{2} B'' y(x)</math>

<math display="block">K(x) = -\frac{d}{dx} (A(x) y(x)) + n A' y(x)</math>

<math display="block">L(x) = \lambda_n y(x)</math>

Then the equation simplifies to <math>\frac{d^n}{dx^n} (J+K + L) = 0</math>

<math>J(x)</math> has three terms, call them in order <math>J_1(x), J_2(x), J_3(x)</math>. <math>K(x)</math> has two terms, call them in order <math>K_1(x), K_2(x)</math>. 

<math>J_3(x) + K_2(x) + L(x) = (\lambda_n + \frac{n(n-1)}{2} B'' + n A')y=0</math>.

That <math>J_1(x) + J_2(x) + K_1(x) = 0</math>. follows from first writing <math>J_1(x)</math> as

<math>J_1(x) = \frac{d^2}{dx^2} \left(B^n(x) \int \exp\left(\frac{A(x)}{B(x)}\right)dx \right)</math>

and then taking the innermost first derivative to obtain

<math>J_1(x) = \frac{d}{dx}\left[\bigg(nB'(x)B^{n-1}(x) + A(x)B^{n-1}(x)\bigg)\int \exp\left(\frac{A(x)}{B(x)}\right)dx\right]</math>

and then rewriting this as

<math>J_1(x) = \frac{d}{dx}\Big(nB'(x)B^{n}(x)w(x)+ A(x)B^{n}(x)w(x)\Big)</math>

The first term is the negative of <math>J_2(x)</math> and the second term is the negative of <math>K_1(x)</math>.
}}

More abstractly, this can be viewed through [[Sturm–Liouville theory]]. Define an operator <math>Lf := - \frac{1}{w} (Wf')'</math>, then the differential equation is equivalent to <math>LP_n = \lambda_n P_n</math>. Define the functional space <math>X = L^2([a,b], w(x)dx)</math> as the Hilbert space of functions over <math>[a, b]</math>, such that <math>\langle f, g\rangle := \int_a^b fgw</math>. Then the operator <math>L</math> is self-adjoint on functions satisfying certain boundary conditions, allowing us to apply the [[spectral theorem]].

== Generating function ==
A simple argument using [[Cauchy's integral formula]] shows that the orthogonal polynomials obtained from the Rodrigues formula have a [[generating function]] of the form
 
<math display="”block”">G(x,u)=\sum_{n=0}^\infty u^nP_n(x)</math> 

The <math>P_n(x)</math> functions here may not have the standard normalizations. But we can write this equivalently as
 
<math display="”block”">G(x,u)=\sum_{n=0}^\infty \frac{u^n}{N_n}N_nP_n(x)</math> 

where the <math>N_n</math> are chosen according to the application so as to give the desired normalizations.  The variable u may be replaced by a constant multiple of u so that

<math display="”block”">G(x,\alpha u)=\sum_{n=0}^\infty \frac{\alpha^n u^n}{N_n}N_nP_n(x)</math> 

This gives an alternate form of the generating function.

By [[Cauchy's integral formula]], Rodrigues’ formula is equivalent to<math display="block">P_n(x)=\frac{n!}{2\pi i}\frac{c_n}{w(x)}\oint_C \frac{B^n(t) w(t)}{(t-x)^{n+1}}\,dt</math>where the integral is along a counterclockwise closed loop around <math>x</math>. Let

<math display=”block”>u=\frac{t-x}{B(t)}</math>

Then the complex path integral takes the form

<math display=”block”>P_n(x)=\frac{n!}{2\pi i}c_n\oint_C \frac{G(x,u)}{u^{n+1}}\,du</math>

<math display=”block”>G(x,u)=\frac{w(t)\frac{dt}{du}}{w(x)B(t)}</math>

where now the closed path C encircles the origin.  In the equation for <math>G(x,u)</math>, <math>t</math> is an implicit function of <math>u</math>. Expanding <math>G(x,u)</math> in the power series given earlier gives

<math>\frac{1}{2\pi i}\oint_C \frac{G(x,u)}{u^{n+1}}\,du=\frac{1}{2\pi i}\oint_C \frac{\sum_{m=0}^\infty u^mP_m(x)}{u^{n+1}}\,du=P_n(x)</math>

Only the <math>m=n</math> term has a nonzero residue, which is <math>P_n(x)</math>.  The <math>n!\,c_n</math> coefficient was dropped since normalizations are conventions which can be inserted afterwards as discussed earlier.

By expressing t in terms of u in the general formula just given for <math>G(x,u)</math>, explicit formulas for <math>G(x,u)</math> may be found.  As a simple example, let <math>B(x)=1</math> and <math>A(x)=-x</math> (Hermite polynomials) so that <math>w(x)=\exp\left(-\frac{x^2}{2}\right)</math>, <math>t=u+x</math>, <math>w(t)=\exp\left(-\frac{(u+x)^2}{2}\right)</math> and so <math>G(x,u)=\exp\left(-xu-\frac{u^2}{2}\right)</math>.

== 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>.

==References==
{{Reflist}}

*{{Citation | last1=Askey | first1=Richard |authorlink=Richard Askey| editor1-last=Altmann | editor1-first=Simón L. | editor2-last=Ortiz | editor2-first=Eduardo L. | title=Mathematics and social utopias in France: Olinde Rodrigues and his times | chapter-url=https://books.google.com/books?id=oTyJYUx8Jr4C&pg=PA105 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series= History of mathematics | isbn=978-0-8218-3860-0 | year=2005 | volume=28 | chapter=The 1839 paper on permutations: its relation to the Rodrigues formula and further developments | pages=105–118}}
*{{citation | title = On the Figure Requisite to Maintain the Equilibrium of a Homogeneous Fluid Mass That Revolves Upon an Axis | last=Ivory|first= James | journal = Philosophical Transactions of the Royal Society of London | volume = 114 | year=1824 | pages = 85–150 | jstor = 107707 | publisher = The Royal Society | doi=10.1098/rstl.1824.0008 | doi-access = free }}
*{{Citation | last1=Jacobi | first1=C. G. J. | title=Ueber eine besondere Gattung algebraischer Functionen, die aus der Entwicklung der Function (1&nbsp;−&nbsp;2''xz''&nbsp;+&nbsp;''z''<sup>2</sup>)<sup>1/2</sup> entstehen. | language=German | doi=10.1515/crll.1827.2.223 | year=1827 | journal=Journal für die Reine und Angewandte Mathematik | issn=0075-4102 | volume=2 | pages=223–226| s2cid=120291793 | url=https://zenodo.org/record/1448810 }}
*{{MacTutor|id=Rodrigues|title=Olinde Rodrigues}}
*{{citation|first=Olinde|last= Rodrigues|authorlink=Olinde Rodrigues|series=(Thesis for the Faculty of Science of the University of Paris)|title=De l'attraction des sphéroïdes|journal=Correspondence sur l'École Impériale Polytechnique|volume=3|issue=3|year=1816|pages= 361–385|url = https://books.google.com/books?id=dp4AAAAAYAAJ&pg=PA361}}

[[Category:Orthogonal polynomials]]