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 functions which are both rational and orthogonal. A rational Legendre function of degree n is defined as:

$R_n(x) = \frac{\sqrt{2}}{x+1}\,L_n\left(\frac{x-1}{x+1}\right)$

where $L_n(x)$ is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm-Liouville problem:

$(x+1)\partial_x(x\partial_x((x+1)v(x)))+\lambda v(x)=0$

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

$R_{n+1}(x)=\frac{2n+1}{n+1}\,\frac{x-1}{x+1}\,R_n(x)-\frac{n}{n+1}\,R_{n-1}(x)\quad\mathrm{for\,n\ge 1}$

and

$2(2n+1)R_n(x)=(x+1)^2(\partial_x R_{n+1}(x)-\partial_x R_{n-1}(x))+(x+1)(R_{n+1}(x)-R_{n-1}(x))$

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

$\lim_{x\rightarrow \infty}(x+1)R_n(x)=\sqrt{2}$

and

$\lim_{x\rightarrow \infty}x\partial_x((x+1)R_n(x))=0$

Orthogonality

$\int_{0}^\infty R_m(x)\,R_n(x)\,dx=\frac{2}{2n+1}\delta_{nm}$

where $\delta_{nm}$ is the Kronecker delta function.

Particular values

$R_0(x)=1\,$

$R_1(x)=\frac{x-1}{x+1}\,$

$R_2(x)=\frac{x^2-4x+1}{(x+1)^2}\,$

$R_3(x)=\frac{x^3-9x^2+9x-1}{(x+1)^3}\,$

$R_4(x)=\frac{x^4-16x^3+36x^2-16x+1}{(x+1)^4}\,$