# Legendre rational functions

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:

$(x+1)\partial _{x}(x\partial _{x}((x+1)v(x)))+\lambda v(x)=0$ 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

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

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

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

$R_{0}(x)=1\,$
$R_{1}(x)={\frac {x-1}{x+1}}\,$
$R_{2}(x)={\frac {x^{2}-4x+1}{(x+1)^{2}}}\,$
$R_{3}(x)={\frac {x^{3}-9x^{2}+9x-1}{(x+1)^{3}}}\,$
$R_{4}(x)={\frac {x^{4}-16x^{3}+36x^{2}-16x+1}{(x+1)^{4}}}\,$