*This article is ***not** about the Chebyshev rational functions used in the design of elliptic filters. For those functions, see Elliptic 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

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

### Differential equations

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

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

### 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

*x*_{0} is a zero, then 1/

*x*_{0} 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 $n\geq 1$ 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

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

## References

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