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: $${\displaystyle R_{n}(x)={\frac {\sqrt {2}}{x+1}}\,P_{n}\left({\frac {x-1}{x+1}}\right)}$$ where ${\displaystyle P_{n}(x)}$ is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm–Liouville problem: $${\displaystyle (x+1){\frac {d}{dx}}\left(x{\frac {d}{dx}}\left[\left(x+1\right)v(x)\right]\right)+\lambda v(x)=0}$$ with eigenvalues $${\displaystyle \lambda _{n}=n(n+1)\,}$$