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.

Plot of the Legendre rational functions for n=0,1,2 and 3 for x between 0.01 and 100.

A rational Legendre function of degree n is defined as:

${\displaystyle R_{n}(x)={\frac {\sqrt {2}}{x+1}}\,P_{n}\left({\frac {x-1}{x+1}}\right)}$

where ${\displaystyle P_{n}(x)}$ is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm-Liouville problem:

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

with eigenvalues

${\displaystyle \lambda _{n}=n(n+1)\,}$

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

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

and

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

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

${\displaystyle \lim _{x\rightarrow \infty }(x+1)R_{n}(x)={\sqrt {2}}}$

and

${\displaystyle \lim _{x\rightarrow \infty }x\partial _{x}((x+1)R_{n}(x))=0}$

Orthogonality

${\displaystyle \int _{0}^{\infty }R_{m}(x)\,R_{n}(x)\,dx={\frac {2}{2n+1}}\delta _{nm}}$

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

Particular values

${\displaystyle R_{0}(x)=1\,}$

${\displaystyle R_{1}(x)={\frac {x-1}{x+1}}\,}$

${\displaystyle R_{2}(x)={\frac {x^{2}-4x+1}{(x+1)^{2}}}\,}$

${\displaystyle R_{3}(x)={\frac {x^{3}-9x^{2}+9x-1}{(x+1)^{3}}}\,}$

${\displaystyle R_{4}(x)={\frac {x^{4}-16x^{3}+36x^{2}-16x+1}{(x+1)^{4}}}\,}$

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.