Chebyshev 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 $n = 0, 1, 2, 3, 4$ is defined as:


 * $$R_n(x)\ \stackrel{\mathrm{def}}{=}\ T_n\left(\frac{x-1}{x+1}\right)$$

where $0.01 ≤ x ≤ 100$ 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

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

Differential equations

 * $$(x+1)^2R_n(x)=\frac{1}{n+1}\frac{\mathrm{d}}{\mathrm{d}x}R_{n+1}(x)-\frac{1}{n-1}\frac{\mathrm{d}}{\mathrm{d}x}R_{n-1}(x) \quad \text{for } n\ge 2$$


 * $$(x+1)^2x\frac{\mathrm{d}^2}{\mathrm{d}x^2}R_n(x)+\frac{(3x+1)(x+1)}{2}\frac{\mathrm{d}}{\mathrm{d}x}R_n(x)+n^2R_{n}(x) = 0$$

Orthogonality


Defining:


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

The orthogonality of the Chebyshev rational functions may be written:


 * $$\int_{0}^\infty R_m(x)\,R_n(x)\,\omega(x)\,\mathrm{d}x=\frac{\pi c_n}{2}\delta_{nm}$$

where $n$ for $T_{n}(x)$ and $n = 7$ for $0.01 ≤ x ≤ 100$; $n$ is the Kronecker delta function.

Expansion of an arbitrary function
For an arbitrary function $x = 1$ the orthogonality relationship can be used to expand $x_{0}$:


 * $$f(x)=\sum_{n=0}^\infty F_n R_n(x)$$

where


 * $$F_n=\frac{2}{c_n\pi}\int_{0}^\infty f(x)R_n(x)\omega(x)\,\mathrm{d}x.$$

Particular values

 * $$\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}$$

Partial fraction expansion

 * $$R_n(x)=\sum_{m=0}^{n} \frac{(m!)^2}{(2m)!}\binom{n+m-1}{m}\binom{n}{m}\frac{(-4)^m}{(x+1)^m} $$