# Chebyshev rational functions

Plot of the Chebyshev rational functions for n = 0, 1, 2, 3 and 4 for x between 0.01 and 100.

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 n is defined as:

$R_n(x)\ \stackrel{\mathrm{def}}{=}\ T_n\left(\frac{x-1}{x+1}\right)$

where $T_n(x)$ 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

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

### Differential equations

$(x+1)^2R_n(x)=\frac{1}{n+1}\frac{d}{dx}\,R_{n+1}(x)-\frac{1}{n-1}\frac{d}{dx}\,R_{n-1}(x) \quad\mathrm{for\,n\ge 2}$
$(x+1)^2x\frac{d^2}{dx^2}\,R_n(x)+\frac{(3x+1)(x+1)}{2}\frac{d}{dx}\,R_n(x)+n^2R_{n}(x) = 0$

### Orthogonality

Plot of the absolute value of the seventh order (n = 7) Chebyshev rational function 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. The maximum value between the zeros is unity. These properties hold for all orders.

Defining:

$\omega(x) \ \stackrel{\mathrm{def}}{=}\ \frac{1}{(x+1)\sqrt{x}}$

The orthogonality of the Chebyshev rational functions may be written:

$\int_{0}^\infty R_m(x)\,R_n(x)\,\omega(x)\,dx=\frac{\pi c_n}{2}\delta_{nm}$

where $c_n$ equals 2 for n = 0 and $c_n$ equals 1 for $n \ge 1$ and $\delta_{nm}$ is the Kronecker delta function.

### Expansion of an arbitrary function

For an arbitrary function $f(x)\in L_\omega^2$ the orthogonality relationship can be used to expand $f(x)$:

$f(x)=\sum_{n=0}^\infty F_n R_n(x)$

where

$F_n=\frac{2}{c_n\pi}\int_{0}^\infty f(x)R_n(x)\omega(x)\,dx.$

## Particular values

$R_0(x)=1\,$
$R_1(x)=\frac{x-1}{x+1}\,$
$R_2(x)=\frac{x^2-6x+1}{(x+1)^2}\,$
$R_3(x)=\frac{x^3-15x^2+15x-1}{(x+1)^3}\,$
$R_4(x)=\frac{x^4-28x^3+70x^2-28x+1}{(x+1)^4}\,$
$R_n(x)=\frac{1}{(x+1)^n}\sum_{m=0}^{n} (-1)^m{2n \choose 2m}x^{n-m}\,$

## Partial fraction expansion

$R_n(x)=\sum_{m=0}^{n} \frac{(m!)^2}{(2m)!}{n+m-1 \choose m}{n \choose m}\frac{(-4)^m}{(x+1)^m}$