Chebyshev rational functions

File:ChebychevRational1.png

Plot of the Chebyshev rational functions for Template:Math for Template:Math, log scale.

In mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev. A rational Chebyshev function of degree Template:Math is defined as:

$R_n(x)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{T}_n\left(\frac{x-1}{x+1}\right)$

where Template:Math is a Chebyshev polynomial of the first kind.

Properties

Many properties can be derived from the properties of the Chebyshev polynomials of the first kind. Other properties are unique to the functions themselves.

Recursion

$R_{n+1}(x)=2\left(\frac{x-1}{x+1}\right)R_n(x)-R_{n-1}(x)\text{for}n\ge 1$

Differential equations

$(x+1{)}^2{R}_n(x)=\frac{1}{n+1}\frac{\mathrm{d}}{\mathrm{d}x}{R}_{n+1}(x)-\frac{1}{n-1}\frac{\mathrm{d}}{\mathrm{d}x}{R}_{n-1}(x)\text{for }n\ge 2$

$(x+1{)}^{2}x\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}{R}_n(x)+\frac{(3x+1)(x+1)}{2}\frac{\mathrm{d}}{\mathrm{d}x}{R}_n(x)+n^2{R}_n(x)=0$

Orthogonality

File:ChebychevRational2.png

Plot of the absolute value of the seventh-order (Template:Math) Chebyshev rational function for Template:Math. Note that there are Template:Math zeroes arranged symmetrically about Template:Math and if Template:Math is a zero, then Template:Math is a zero as well. The maximum value between the zeros is unity. These properties hold for all orders.

Defining:

$\omega (x)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{(x+1)\sqrt{x}}$

The orthogonality of the Chebyshev rational functions may be written:

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{R}_m(x)R_n(x)\omega (x)\mathrm{d}x=\frac{\pi c_n}{2}{\delta }_{nm}$

where Template:Math for Template:Math and Template:Math for Template:Math; Template:Math is the Kronecker delta function.

Expansion of an arbitrary function

For an arbitrary function Template:Math the orthogonality relationship can be used to expand Template:Math:

$f(x)=\sum _{n=0}^{\mathrm{\infty }}{F}_n{R}_n(x)$

where

$F_n=\frac{2}{c_n\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}f(x)R_n(x)\omega (x)\mathrm{d}x.$

Particular values

$\begin{array}{rl}{R}_0(x)& =1\\ R_1(x)& =\frac{x-1}{x+1}\\ R_2(x)& =\frac{x^2-6x+1}{(x+1{)}^2}\\ R_3(x)& =\frac{x^3-15x^2+15x-1}{(x+1{)}^3}\\ R_4(x)& =\frac{x^4-28x^3+70x^2-28x+1}{(x+1{)}^4}\\ R_n(x)& =(x+1{)}^{-n}\sum _{m=0}^n(-1{)}^m(\genfrac{}{}{0ex}{}{2n}{2m})x^{n-m}\end{array}$

Partial fraction expansion

$R_n(x)=\sum _{m=0}^n\frac{(m!{)}^2}{(2m)!}(\genfrac{}{}{0ex}{}{n+m-1}{m})(\genfrac{}{}{0ex}{}{n}{m})\frac{(-4{)}^m}{(x+1{)}^m}$