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:

x + 1

(

)

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

{\displaystyle P_{n}(x)}

is a Legendre polynomial.

These functions are eigenfunctions of the singular Sturm–Liouville problem:

( x + 1 )

+ λ v ( x ) = 0

{\displaystyle (x+1){\frac {d}{dx}}\left(x{\frac {d}{dx}}\left[\left(x+1\right)v(x)\right]\right)+\lambda v(x)=0}

with eigenvalues

Many properties can be derived from the properties of the Legendre polynomials of the first kind.

Other properties are unique to the functions themselves.

{\displaystyle 2(2n+1)R_{n}(x)=\left(x+1\right)^{2}\left({\frac {d}{dx}}R_{n+1}(x)-{\frac {d}{dx}}R_{n-1}(x)\right)+(x+1)\left(R_{n+1}(x)-R_{n-1}(x)\right)}

It can be shown that

lim

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

{\displaystyle \delta _{nm}}

is the Kronecker delta function.

{\displaystyle {\begin{aligned}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{aligned}}}