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: $$(x+1) \frac{d}{dx}\left(x \frac{d}{dx} \left[\left(x+1\right) v(x)\right]\right) + \lambda v(x) = 0$$ 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 $$2 (2n+1) R_n(x) = \left(x+1\right)^2 \left(\frac{d}{dx} R_{n+1}(x) - \frac{d}{dx} R_{n-1}(x)\right) + (x+1) \left(R_{n+1}(x) - R_{n-1}(x)\right)$$

Limiting behavior
It can be shown that $$\lim_{x\to\infty}(x+1)R_n(x)=\sqrt{2}$$ and $$\lim_{x\to\infty}x\partial_x((x+1)R_n(x))=0$$

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
$$\begin{align} R_0(x) &= \frac{\sqrt{2}}{x+1}\,1 \\ R_1(x) &= \frac{\sqrt{2}}{x+1}\,\frac{x-1}{x+1} \\ R_2(x) &= \frac{\sqrt{2}}{x+1}\,\frac{x^2-4x+1}{(x+1)^2} \\ R_3(x) &= \frac{\sqrt{2}}{x+1}\,\frac{x^3-9x^2+9x-1}{(x+1)^3} \\ R_4(x) &= \frac{\sqrt{2}}{x+1}\,\frac{x^4-16x^3+36x^2-16x+1}{(x+1)^4} \end{align}$$