Chebyshev rational functions

${\displaystyle R_n(x)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{T}_n\left(\frac{x-1}{x+1}\right)}$

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

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

Differential equations

${\displaystyle (x+1{)}^2{R}_n(x)=\frac{1}{n+1}\frac{\mathrm{d}}{\mathrm{d}x}{R}_{n+1}(x)-\frac{1}{n-1}\frac{\mathrm{d}}{\mathrm{d}x}{R}_{n-1}(x)\text{for }n\ge 2}$

${\displaystyle (x+1{)}^{2}x\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}{R}_n(x)+\frac{(3x+1)(x+1)}{2}\frac{\mathrm{d}}{\mathrm{d}x}{R}_n(x)+n^2{R}_n(x)=0}$

Orthogonality

Defining:

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

The orthogonality of the Chebyshev rational functions may be written:

${\displaystyle {\int }_0^{\mathrm{\infty }}{R}_m(x)R_n(x)\omega (x)\mathrm{d}x=\frac{\pi c_n}{2}{\delta }_{nm}}$

where c_n = 2 for n = 0 and c_n = 1 for n ≥ 1; δ_nm 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):

${\displaystyle f(x)=\sum _{n=0}^{\mathrm{\infty }}{F}_n{R}_n(x)}$

where

${\displaystyle F_n=\frac{2}{c_n\pi }{\int }_0^{\mathrm{\infty }}f(x)R_n(x)\omega (x)\mathrm{d}x.}$