Legendre rational functions

Short description: Sequence of orthogonal functions on [0, ∞)

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:

$(x+1){\partial }_x(x{\partial }_x((x+1)v(x)))+\lambda v(x)=0$

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

$R_{n+1}(x)=\frac{2n+1}{n+1}\frac{x-1}{x+1}{R}_n(x)-\frac{n}{n+1}{R}_{n-1}(x)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{\ge }\mathrm{1}$

and

$2(2n+1)R_n(x)=(x+1{)}^2({\partial }_x{R}_{n+1}(x)-{\partial }_x{R}_{n-1}(x))+(x+1)(R_{n+1}(x)-R_{n-1}(x))$

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

$\underset{x\to \mathrm{\infty }}{\lim }(x+1)R_n(x)=\sqrt{2}$

and

$\underset{x\to \mathrm{\infty }}{\lim }x{\partial }_x((x+1)R_n(x))=0$

Orthogonality

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{R}_m(x)R_n(x)dx=\frac{2}{2n+1}{\delta }_{nm}$

where ${\delta }_{nm}$ is the Kronecker delta function.

Particular values

$R_0(x)=\frac{\sqrt{2}}{x+1}1$

$R_1(x)=\frac{\sqrt{2}}{x+1}\frac{x-1}{x+1}$

$R_2(x)=\frac{\sqrt{2}}{x+1}\frac{x^2-4x+1}{(x+1{)}^2}$

$R_3(x)=\frac{\sqrt{2}}{x+1}\frac{x^3-9x^2+9x-1}{(x+1{)}^3}$

$R_4(x)=\frac{\sqrt{2}}{x+1}\frac{x^4-16x^3+36x^2-16x+1}{(x+1{)}^4}$

References

Zhong-Qing, Wang; Ben-Yu, Guo (2005). "A mixed spectral method for incompressible viscous fluid flow in an infinite strip". Mat. Apl. Comput. 24 (3). doi:10.1590/S0101-82052005000300002.