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:

${\displaystyle R_{n}(x)={\frac {\sqrt {2)){x+1))\,P_{n}\left({\frac {x-1}{x+1))\right)}$

where ${\displaystyle P_{n}(x)}$ is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm–Liouville problem:

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

with eigenvalues

${\displaystyle \lambda _{n}=n(n+1)\,}$

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

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

and

${\displaystyle 2(2n+1)R_{n}(x)=(x+1)^{2}(\partial _{x}R_{n+1}(x)-\partial _{x}R_{n-1}(x))+(x+1)(R_{n+1}(x)-R_{n-1}(x))}$

Limiting behavior

It can be shown that

${\displaystyle \lim _{x\rightarrow \infty }(x+1)R_{n}(x)={\sqrt {2))}$

and

${\displaystyle \lim _{x\rightarrow \infty }x\partial _{x}((x+1)R_{n}(x))=0}$

Orthogonality

${\displaystyle \int _{0}^{\infty }R_{m}(x)\,R_{n}(x)\,dx={\frac {2}{2n+1))\delta _{nm))$

where ${\displaystyle \delta _{nm))$ is the Kronecker delta function.

Particular values

${\displaystyle R_{0}(x)={\frac {\sqrt {2)){x+1))\,1\,}$

${\displaystyle R_{1}(x)={\frac {\sqrt {2)){x+1))\,{\frac {x-1}{x+1))\,}$

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

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

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