Legendre rational functions

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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 functions which are both rational and orthogonal. A rational Legendre function of degree n is defined as:

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

where $L n (x)$ 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

$\lambda_n=n(n+1)\,$

1 Properties
2 Particular values
3 References

[edit] 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.

[edit] Recursion

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

and

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

[edit] 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

$\lim_{x\rightarrow \infty}(x+1)R_n(x)=\sqrt{2}$