Legendre rational functions

Plot of the Legendre rational functions for n=0,1,2 and 3 for x between 0.01 and 100.

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

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

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

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

References

• Zhong-Qing, Wang; Ben-Yu, Guo (2005). "A mixed spectral method for incompressible viscous fluid flow in an infinite strip". Computational & Applied Mathematics. 24 (3). Sociedade Brasileira de Matemática Aplicada e Computacional. doi:10.1590/S0101-82052005000300002.