Legendre rational functions

File:LegendreRational1.png

Plot of the Legendre rational functions for n=0,1,2 and 3 for x between 0.01 and 100.

In mathematics, the Legendre rational functions are a sequence of orthogonal functions on [0, ∞). They are obtained by composing the Cayley transform with Legendre polynomials.

A rational Legendre function of degree n is defined as: $$R_n(x)=\frac{\sqrt{2}}{x+1}{P}_n\left(\frac{x-1}{x+1}\right)$$ where $P_n(x)$ is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm–Liouville problem: $$(x+1)\frac{d}{dx}\left(x\frac{d}{dx}[(x+1)v(x)]\right)+\lambda v(x)=0$$ with eigenvalues $${\lambda }_n=n(n+1)$$

Properties

Many properties can be derived from the properties of the Legendre polynomials of the first kind. Other properties are unique to the functions themselves.

Recursion

$$R_{n+1}(x)=\frac{2n+1}{n+1}\frac{x-1}{x+1}{R}_n(x)-\frac{n}{n+1}{R}_{n-1}(x)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{\ge }\mathrm{1}$$ and $$2(2n+1)R_n(x)={(x+1)}^2(\frac{d}{dx}{R}_{n+1}(x)-\frac{d}{dx}{R}_{n-1}(x))+(x+1)(R_{n+1}(x)-R_{n-1}(x))$$

Limiting behavior

File:LegendreRational2.png

Plot of the seventh order (n=7) Legendre rational function multiplied by 1+x for x between 0.01 and 100. Note that there are n zeroes arranged symmetrically about x=1 and if x₀ is a zero, then 1/x₀ is a zero as well. These properties hold for all orders.

It can be shown that $$\underset{x\to \mathrm{\infty }}{\lim }(x+1)R_n(x)=\sqrt{2}$$ and $$\underset{x\to \mathrm{\infty }}{\lim }x{\partial }_x((x+1)R_n(x))=0$$

Orthogonality

$$\underset{0}{\overset{\mathrm{\infty }}{\int }}{R}_m(x)R_n(x)dx=\frac{2}{2n+1}{\delta }_{nm}$$ where ${\delta }_{nm}$ is the Kronecker delta function.

Particular values

$$\begin{array}{cc}{R}_0(x)& =\frac{\sqrt{2}}{x+1}1\\ R_1(x)& =\frac{\sqrt{2}}{x+1}\frac{x-1}{x+1}\\ R_2(x)& =\frac{\sqrt{2}}{x+1}\frac{x^2-4x+1}{(x+1{)}^2}\\ R_3(x)& =\frac{\sqrt{2}}{x+1}\frac{x^3-9x^2+9x-1}{(x+1{)}^3}\\ R_4(x)& =\frac{\sqrt{2}}{x+1}\frac{x^4-16x^3+36x^2-16x+1}{(x+1{)}^4}\end{array}$$