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

Many properties can be derived from the properties of the Legendre polynomials of the first kind. Other properties are unique to the functions themselves.

$${\displaystyle R_{n+1}(x)={\frac {2n+1}{n+1}},円{\frac {x-1}{x+1}},円R_{n}(x)-{\frac {n}{n+1}},円R_{n-1}(x)\quad \mathrm {for,円n\geq 1} }$${\displaystyle R_{n+1}(x)={\frac {2n+1}{n+1}},円{\frac {x-1}{x+1}},円R_{n}(x)-{\frac {n}{n+1}},円R_{n-1}(x)\quad \mathrm {for,円n\geq 1} } and $${\displaystyle 2(2n+1)R_{n}(x)=\left(x+1\right)^{2}\left({\frac {d}{dx}}R_{n+1}(x)-{\frac {d}{dx}}R_{n-1}(x)\right)+(x+1)\left(R_{n+1}(x)-R_{n-1}(x)\right)}$${\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 $${\displaystyle \lim _{x\to \infty }(x+1)R_{n}(x)={\sqrt {2}}}$${\displaystyle \lim _{x\to \infty }(x+1)R_{n}(x)={\sqrt {2}}} and $${\displaystyle \lim _{x\to \infty }x\partial _{x}((x+1)R_{n}(x))=0}$${\displaystyle \lim _{x\to \infty }x\partial _{x}((x+1)R_{n}(x))=0}

$${\displaystyle \int _{0}^{\infty }R_{m}(x),円R_{n}(x),円dx={\frac {2}{2n+1}}\delta _{nm}}$${\displaystyle \int _{0}^{\infty }R_{m}(x),円R_{n}(x),円dx={\frac {2}{2n+1}}\delta _{nm}} where ${\displaystyle \delta _{nm}}${\displaystyle \delta _{nm}} is the Kronecker delta function.

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

• 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 .

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