Chebyshev rational functions

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This article is not about the Chebyshev rational functions used in the design of elliptic filters. For these 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. 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.

1 Properties
2 Particular values
3 References

[edit] 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.

[edit] Recursion

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

[edit] Differential equations

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

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

[edit] 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:

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