Rodrigues' formula

Template:Short description Template:For In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by Template:Harvs, Template:Harvs and Template:Harvs. The name "Rodrigues formula" was introduced by 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. Template:Harvtxt describes the history of the Rodrigues formula in detail.

Statement

Let $(P_n(x){)}_{n=0}^{\mathrm{\infty }}$ be a sequence of orthogonal polynomials defined on the interval $[a,b]$ satisfying the orthogonality condition $$\underset{a}{\overset{b}{\int }}{P}_m(x)P_n(x)w(x)dx=K_n{\delta }_{m,n},$$ where $w(x)$ is a suitable weight function, $K_n$ is a constant depending on $n$, and ${\delta }_{m,n}$ is the Kronecker delta. The weight function w(x)=W(x)/B(x) where the integration factor W(x) satisfies the equation $$\frac{W^{\prime }(x)}{W(x)}=\frac{A(x)}{B(x)},$$ where $A(x)$ is a polynomial with degree at most 1 and $B(x)$ is a polynomial with degree at most 2. For instance, for the Legendre polynomials, B(x)=1-x*x and A(x)=-2x. This gives W(x)=1-x*x and w(x)=1. Further, the limits $$\underset{x\to a}{\lim }w(x)B(x)=0,\underset{x\to b}{\lim }w(x)B(x)=0.$$ Then it can be shown that $P_n(x)$ satisfies a relation of the form, $$P_n(x)=\frac{c_n}{w(x)}\frac{d^n}{d{x}^n}[B(x{)}^{n}w(x)],$$ for some constants $c_n$. This relation is called Rodrigues' type formula, or just Rodrigues' formula.

Polynomials obtained from Rodrigues' formula obey the second order differential equation for the classical orthogonal polynomials

$B(x)\frac{d^2}{d{x}^2}{P}_n(x)+A(x)\frac{d}{dx}{P}_n(x)+{\lambda }_n{P}_n(x)=0$

${\lambda }_n=-\frac{1}{2}n(n-1)B^{″}-n{A}^{\prime }$

The second derivative of B(x) and first derivative of A(x) are constants. Continuing the example of the Legendre polynomials,

${\lambda }_n=n(n+1)$

Differential equation proof

The following proof shows that the polynomials obtained from the Rodrigues' formula obey the second order differential equation just given. This proof repeatedly uses the fact that the second derivative of B(x) and the first derivative of A(x) are constants. Using

$\frac{1}{w(x)}=B(x)\exp (-\int \frac{A(x)}{B(x)}dx)$

the differential equation that we are to prove may be put in the form

$B(x)\frac{d^2}{d{x}^2}{I}_n(x)+(2B^{\prime }(x)-A(x))\frac{d}{dx}{I}_n(x)+(2B^{″}-A^{\prime }+{\lambda }_n)I_n(x)=0$

$I_n(x)=\frac{d^n}{d{x}^n}(B^n(x)w(x))$

This is equivalent to

$\frac{d^2}{d{x}^2}(B(x)I_n(x))-\frac{d}{dx}(A(x)I_n(x))+{\lambda }_n{I}_n(x)=0$

This is the differential equation that we will prove to be true. We will do so using the following two identities that move B(x) and A(x) to the other side of the $n^{th}$ derivative

$B(x)\frac{d^n}{d{x}^n}y=\frac{d^n}{d{x}^n}(B(x)y)-n\frac{d^{n-1}}{d{x}^{n-1}}(B^{\prime }(x)y)+\frac{n(n-1)}{2}\frac{d^{n-2}}{d{x}^{n-2}}(B^{″}y)$

$A(x)\frac{d^n}{d{x}^n}y=\frac{d^n}{d{x}^n}(A(x)y)-n\frac{d^{n-1}}{d{x}^{n-1}}(A^{\prime }y)$

$y=B^n(x)w(x)$

The second order, first order, and zero-th order derivatives have the respective forms $\frac{d^n}{d{x}^n}J(x)$, $\frac{d^n}{d{x}^n}K(x)$, $\frac{d^n}{d{x}^n}L(x)$.

$J(x)=\frac{d^2}{d{x}^2}(B^{n+1}(x)w(x))-n\frac{d}{dx}(B^{\prime }(x)B^n(x)w(x))+\frac{n(n-1)}{2}(B^{″}{B}^n(x)w(x))$

$K(x)=-\frac{d}{dx}(A(x)B^n(x)w(x))+n{A}^{\prime }{B}^n(x)w(x)$

$L(x)={\lambda }_n{B}^n(x)w(x)$

$J(x)$ has three terms, call them in order $J_1(x)$, $J_2(x)$, and $J_3(x)$. $K(x)$ has two terms, call them in order $K_1(x)$ and $K_2(x)$. The sum $J_1(x)+J_2(x)+K_1(x)$ is zero. The sum $J_3(x)+K_2(x)+L(x)$ is zero provided that ${\lambda }_n$ is given by the equation given earlier. Done.

Generating functions

Rodrigues’ formula together with Cauchy’s Residue theorem for complex integration on a closed path enclosing poles gives the generating functions having the property $G(x,u)=\sum _{n=0}^{\mathrm{\infty }}{u}^n{P}_n(x)$ Here's how:

By Cauchy’s Residue Theorem, Rodrigues’ formula is equivalent to

$P_n(x)=\frac{n!}{2\pi i}\frac{1}{w(x)}{\oint }_C\frac{B^n(t)w(t)}{(t-x{)}^{n+1}}dt$

where the complex variable t is integrated along a counterclockwise closed path C that encircles x. Make the change of variable

$u=\frac{t-x}{B(t)}$

Then the complex path integral takes the form

$P_n(x)=\frac{n!}{2\pi i}{\oint }_C\frac{G(x,u)}{u^{n+1}}du$

$G(x,u)=\frac{w(t)\frac{dt}{du}}{w(x)B(t)}$

where now the closed path C encircles the origin. In the equation for $G(x,u)$, $t$ is an implicit function of $u$. As examples, we will find the generating functions for the Hermite polynomials and the Legendre polynomials.

The Hermite polynomials are particularly easy:

$B(x)=1$

$A(x)=-x$

$w(x)=\exp (-x^2/2)$

$u=t-x$

$w(t)=\exp (-\frac{(x+u{)}^2}{2})$

$\frac{w(t)}{w(x)}=\exp (-xu-\frac{u^2}{2})$

The generating function is

$\exp (-xu-u^2/2)=\sum _{n=0}^{\mathrm{\infty }}{u}^n{H}_n(x)$

where $H_n(x)$ are the Hermite polynomials. If we replace $u$ by $-u$ and $H_n(x)$ by $(-1{)}^n{H}_n(x)$, we get the usual generating function relationship.

The Legendre polynomials require more work.

$B(x)=x^2-1$

$A(x)=2x$

$w(x)=w(t)=1$

$u=\frac{t-x}{B(t)}$

The last equation implicitly gives $t$ as a function of $u$. We find

$t=\frac{1-\sqrt{R}}{2u}$

$R=4u^2-4ux+1$

Then

$B(t)=\frac{2-4ux-2\sqrt{R}}{4u^2}$

$\frac{dt}{du}=\frac{2-4ux-2\sqrt{R}}{4u^2\sqrt{R}}$

$G(x,u)=\frac{1}{\sqrt{R}}$

If we replace $u$ by $\frac{u}{2}$ and $P_n(x)$ by $2^n{P}_n(x)$, we get the generating function relation in the usual form.

Applications

The most known applications of Rodrigues' type formulas are the formulas for Legendre, Laguerre and Hermite polynomials:

Rodrigues stated his formula for Legendre polynomials $P_n$: $$P_n(x)=\frac{1}{2^{n}n!}\frac{d^n}{d{x}^n}[(x^2-1{)}^n].$$

Laguerre polynomials are usually denoted L₀, L₁, ..., and the Rodrigues formula can be written as $$L_n(x)=\frac{e^x}{n!}\frac{d^n}{d{x}^n}\left[e^{-x}{x}^n\right]=\frac{1}{n!}{(\frac{d}{dx}-1)}^n{x}^n,$$

The Rodrigues formula for the Hermite polynomials can be written as $$H_n(x)=(-1{)}^n{e}^{x^2}\frac{d^n}{d{x}^n}\left[e^{-x^2}\right]={(2x-\frac{d}{dx})}^n\cdot 1.$$

Similar formulae hold for many other sequences of orthogonal functions arising from Sturm–Liouville equations, and these are also called the Rodrigues formula (or Rodrigues' type formula) for that case, especially when the resulting sequence is polynomial.

References

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