Legendre rational functions

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}\,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)\,

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)\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))

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}

and

\lim_{x\rightarrow \infty}x\partial_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

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. Retrieved 2006-08-08.

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