Chebyshev rational functions

Plot of the Chebyshev rational functions for n = 0, 1, 2, 3, 4 for 0.01 ≤ x ≤ 100, 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 n is defined as:

[math]\displaystyle{ R_n(x)\ \stackrel{\mathrm{def}}{=}\ T_n\left(\frac{x-1}{x+1}\right) }[/math]

where Tn(x) is a Chebyshev polynomial of the first kind.

## Contents

* 1 Properties
* 1.1 Recursion
* 1.2 Differential equations
* 1.3 Orthogonality
* 1.4 Expansion of an arbitrary function
* 2 Particular values
* 3 Partial fraction expansion
* 4 References

## 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

[math]\displaystyle{ R_{n+1}(x)=2\left(\frac{x-1}{x+1}\right)R_{n}(x)-R_{n-1}(x)\quad\text{for}\,n\ge 1 }[/math]

### Differential equations

[math]\displaystyle{ (x+1)^2R_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) \quad \text{for } n\ge 2 }[/math]

[math]\displaystyle{ (x+1)^2x\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^2R_{n}(x) = 0 }[/math]

### Orthogonality

Plot of the absolute value of the seventh-order (n = 7) Chebyshev rational function for 0.01 ≤ x ≤ 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:

[math]\displaystyle{ \omega(x) \ \stackrel{\mathrm{def}}{=}\ \frac{1}{(x+1)\sqrt{x}} }[/math]

The orthogonality of the Chebyshev rational functions may be written:

[math]\displaystyle{ \int_{0}^\infty R_m(x)\,R_n(x)\,\omega(x)\,\mathrm{d}x=\frac{\pi c_n}{2}\delta_{nm} }[/math]

where cn = 2 for n = 0 and cn = 1 for n ≥ 1; δnm is the Kronecker delta function.

### Expansion of an arbitrary function

For an arbitrary function f(x) ∈ L2ω the orthogonality relationship can be used to expand f(x):

[math]\displaystyle{ f(x)=\sum_{n=0}^\infty F_n R_n(x) }[/math]

where

[math]\displaystyle{ F_n=\frac{2}{c_n\pi}\int_{0}^\infty f(x)R_n(x)\omega(x)\,\mathrm{d}x. }[/math]

## Particular values

[math]\displaystyle{ \begin{align} 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\binom{2n}{2m}x^{n-m} \end{align} }[/math]

## Partial fraction expansion

[math]\displaystyle{ R_n(x)=\sum_{m=0}^{n} \frac{(m!)^2}{(2m)!}\binom{n+m-1}{m}\binom{n}{m}\frac{(-4)^m}{(x+1)^m} }[/math]

## References

* Guo, Ben-Yu; Shen, Jie; Wang, Zhong-Qing (2002). "Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval". Int. J. Numer. Methods Eng. 53 (1): 65–84. doi:10.1002/nme.392. Bibcode: 2002IJNME..53...65G. http://www.math.purdue.edu/~shen/pub/GSW_IJNME02.pdf. Retrieved 2006-07-25.

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