# Chebyshev rational functions

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff) 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:

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

$\omega (x)\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{(x+1){\sqrt {x}}}}$ The orthogonality of the Chebyshev rational functions may be written:

$\int _{0}^{\infty }R_{m}(x)\,R_{n}(x)\,\omega (x)\,dx={\frac {\pi c_{n}}{2}}\delta _{nm}$ ### Expansion of an arbitrary function

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

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