Chebyshev rational functions

This article is not about the Chebyshev rational functions used in the design of elliptic filters. For those functions, see Elliptic rational functions.
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%2B1}\right)

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

Contents

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%2B1}(x)=2\,\frac{x-1}{x%2B1}R_n(x)-R_{n-1}(x)\quad\mathrm{for\,n\ge 1}

Differential equations

(x%2B1)^2R_n(x)=\frac{1}{n%2B1}\frac{d}{dx}\,R_{n%2B1}(x)-\frac{1}{n-1}\frac{d}{dx}\,R_{n-1}(x) \quad\mathrm{for\,n\ge 2}
(x%2B1)^2x\frac{d^2}{dx^2}\,R_n(x)%2B\frac{(3x%2B1)(x%2B1)}{2}\frac{d}{dx}\,R_n(x)%2Bn^2R_{n}(x) = 0

Orthogonality

Defining:

\omega(x) \ \stackrel{\mathrm{def}}{=}\ \frac{1}{(x%2B1)\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 \ge 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%2B1}\,
R_2(x)=\frac{x^2-6x%2B1}{(x%2B1)^2}\,
R_3(x)=\frac{x^3-15x^2%2B15x-1}{(x%2B1)^3}\,
R_4(x)=\frac{x^4-28x^3%2B70x^2-28x%2B1}{(x%2B1)^4}\,
R_n(x)=\frac{1}{(x%2B1)^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%2Bm-1 \choose m}{n \choose m}\frac{(-4)^m}{(x%2B1)^m}

References