Legendre rational functions

In mathematics the Legendre rational functions are a sequence of functions which are both rational and orthogonal. A rational Legendre function of degree n is defined as:

$R_n(x) = \frac{\sqrt{2}}{x%2B1}\,L_n\left(\frac{x-1}{x%2B1}\right)$

where $L_n(x)$ is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm-Liouville problem:

$(x%2B1)\partial_x(x\partial_x((x%2B1)v(x)))%2B\lambda v(x)=0$

with eigenvalues

$\lambda_n=n(n%2B1)\,$

Contents

1 Properties
2 Particular values
3 References

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

$R_{n%2B1}(x)=\frac{2n%2B1}{n%2B1}\,\frac{x-1}{x%2B1}\,R_n(x)-\frac{n}{n%2B1}\,R_{n-1}(x)\quad\mathrm{for\,n\ge 1}$

and

$2(2n%2B1)R_n(x)=(x%2B1)^2(\partial_x R_{n%2B1}(x)-\partial_x R_{n-1}(x))%2B(x%2B1)(R_{n%2B1}(x)-R_{n-1}(x))$

Limiting behavior

It can be shown that

$\lim_{x\rightarrow \infty}(x%2B1)R_n(x)=\sqrt{2}$

and

$\lim_{x\rightarrow \infty}x\partial_x((x%2B1)R_n(x))=0$

Orthogonality

$\int_{0}^\infty R_m(x)\,R_n(x)\,dx=\frac{2}{2n%2B1}\delta_{nm}$

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

Particular values

$R_0(x)=1\,$

$R_1(x)=\frac{x-1}{x%2B1}\,$

$R_2(x)=\frac{x^2-4x%2B1}{(x%2B1)^2}\,$

$R_3(x)=\frac{x^3-9x^2%2B9x-1}{(x%2B1)^3}\,$

$R_4(x)=\frac{x^4-16x^3%2B36x^2-16x%2B1}{(x%2B1)^4}\,$

References

Zhong-Qing, Wang; Ben-Yu, Guo (2005). "A mixed spectral method for incompressible viscous fluid flow in an infinite strip" (PDF). Mat. apl. comput. 24 (3). doi:10.1590/S0101-82052005000300002. http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0101-82052005000300002&lng=en&nrm=iso. Retrieved 2006-08-08.