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 Tn(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

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

### Differential equations

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

### Orthogonality

Defining:

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

The orthogonality of the Chebyshev rational functions may be written:

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

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) ∈ L2
ω
the orthogonality relationship can be used to expand f(x):

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

where

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

## Particular values

{\displaystyle {\begin{aligned}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{aligned))}

## Partial fraction expansion

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

## References

• Guo, Ben-Yu; Shen, Jie; Wang, Zhong-Qing (2002). "Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval" (PDF). Int. J. Numer. Methods Eng. 53 (1): 65–84. Bibcode:2002IJNME..53...65G. CiteSeerX 10.1.1.121.6069. doi:10.1002/nme.392. S2CID 9208244. Retrieved 2006-07-25.