# 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:

${\displaystyle R_{n}(x)\ {\stackrel {\mathrm {def} }{=}}\ T_{n}\left({\frac {x-1}{x+1}}\right)}$

where ${\displaystyle T_{n}(x)}$ 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

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

### Differential equations

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

### 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:

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

The orthogonality of the Chebyshev rational functions may be written:

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

### Expansion of an arbitrary function

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

${\displaystyle f(x)=\sum _{n=0}^{\infty }F_{n}R_{n}(x)}$

where

${\displaystyle F_{n}={\frac {2}{c_{n}\pi }}\int _{0}^{\infty }f(x)R_{n}(x)\omega (x)\,dx.}$

## Particular values

${\displaystyle R_{0}(x)=1\,}$
${\displaystyle R_{1}(x)={\frac {x-1}{x+1}}\,}$
${\displaystyle R_{2}(x)={\frac {x^{2}-6x+1}{(x+1)^{2}}}\,}$
${\displaystyle R_{3}(x)={\frac {x^{3}-15x^{2}+15x-1}{(x+1)^{3}}}\,}$
${\displaystyle R_{4}(x)={\frac {x^{4}-28x^{3}+70x^{2}-28x+1}{(x+1)^{4}}}\,}$
${\displaystyle R_{n}(x)={\frac {1}{(x+1)^{n}}}\sum _{m=0}^{n}(-1)^{m}{2n \choose 2m}x^{n-m}\,}$

## Partial fraction expansion

${\displaystyle R_{n}(x)=\sum _{m=0}^{n}{\frac {(m!)^{2}}{(2m)!}}{n+m-1 \choose m}{n \choose m}{\frac {(-4)^{m}}{(x+1)^{m}}}}$

## References

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