Chebyshev rational functions

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This article is not about the Chebyshev rational functions used in the design of elliptic filters. For those functions, see Elliptic rational functions.
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)

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


Many properties can be derived from the properties of the Chebyshev polynomials of the first kind. Other properties are unique to the functions themselves.


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

Differential equations[edit]

\quad\mathrm{for\,n\ge 2}
(x+1)^2x\frac{d^2}{dx^2}\,R_n(x)+\frac{(3x+1)(x+1)}{2}\frac{d}{dx}\,R_n(x)+n^2R_{n}(x) = 0


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.


\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}

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[edit]

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)


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

Particular values[edit]

R_n(x)=\frac{1}{(x+1)^n}\sum_{m=0}^{n} (-1)^m{2n \choose 2m}x^{n-m}\,

Partial fraction expansion[edit]

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}