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:

:$R\_n(x)\; =\; frac\{sqrt\{2\{x+1\},L\_nleft(frac\{x-1\}\{x+1\}\; ight)$

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

:$(x+1)partial\_x(xpartial\_x((x+1)v(x)))+lambda\; v(x)=0$

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

:$R\_\{n+1\}(x)=frac\{2n+1\}\{n+1\},frac\{x-1\}\{x+1\},R\_n(x)-frac\{n\}\{n+1\},R\_\{n-1\}(x)quadmathrm\{for,nge\; 1\}$

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

It can be shown that

:$lim\_\{x\; ightarrow\; infty\}(x+1)R\_n(x)=sqrt\{2\}$

and

:$lim\_\{x\; ightarrow\; infty\}xpartial\_x((x+1)R\_n(x))=0$

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

:$R\_0(x)=1,$:$R\_1(x)=frac\{x-1\}\{x+1\},$:$R\_2(x)=frac\{x^2-4x+1\}\{(x+1)^2\},$:$R\_3(x)=frac\{x^3-9x^2+9x-1\}\{(x+1)^3\},$:$R\_4(x)=frac\{x^4-16x^3+36x^2-16x+1\}\{(x+1)^4\},$

References

cite journal
last = Zhong-Qing
first = Wang
authorlink =
coauthors = Ben-Yu, Guo
year = 2005
month =
title = A mixed spectral method for incompressible viscous fluid flow in an infinite strip
journal = Mat. apl. comput.
volume = 24
issue = 3
pages =
doi = 10.1590/S0101-82052005000300002
id =
url = http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0101-82052005000300002&lng=en&nrm=iso
format = PDF
accessdate = 2006-08-08