Chebyshev rational functions

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Plot of the Chebyshev rational functions for

n = 0, 1, 2, 3, 4

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0.01 \leq x \leq 100

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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 $n$ Script error: No such module "Check for unknown parameters". is defined as:

R_{n} (x) \overset{d e f}{=} T_{n} (\frac{x - 1}{x + 1})

where $T n (x)$ Script error: No such module "Check for unknown parameters". 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 (\frac{x - 1}{x + 1}) R_{n} (x) - R_{n - 1} (x) for n \geq 1

Differential equations

(x + 1)^{2} R_{n} (x) = \frac{1}{n + 1} \frac{d}{d x} R_{n + 1} (x) - \frac{1}{n - 1} \frac{d}{d x} R_{n - 1} (x) for n \geq 2

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

Orthogonality

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Plot of the absolute value of the seventh-order (

n = 7

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0.01 \leq x \leq 100

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n

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x = 1

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x 0

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Defining:

ω (x) \overset{d e f}{=} \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) ω (x) d x = \frac{π c_{n}}{2} δ_{n m}

where $c n = 2$ Script error: No such module "Check for unknown parameters". for $n = 0$ Script error: No such module "Check for unknown parameters". and $c n = 1$ Script error: No such module "Check for unknown parameters". for $n \geq 1$ Script error: No such module "Check for unknown parameters".; $δ nm$ Script error: No such module "Check for unknown parameters". is the Kronecker delta function.

Expansion of an arbitrary function

For an arbitrary function $f (x) \in L Script error: No such module "Su".$ Script error: No such module "Check for unknown parameters". the orthogonality relationship can be used to expand $f (x)$ Script error: No such module "Check for unknown parameters".:

f (x) = \sum_{n = 0}^{\infty} F_{n} R_{n} (x)

where

F_{n} = \frac{2}{c_{n} π} \int_{0}^{\infty} f (x) R_{n} (x) ω (x) d x .

Particular values

\begin{aligned} R_{0} (x) & = 1 \\ R_{1} (x) & = \frac{x - 1}{x + 1} \\ R_{2} (x) & = \frac{x^{2} - 6 x + 1}{(x + 1)^{2}} \\ R_{3} (x) & = \frac{x^{3} - 15 x^{2} + 15 x - 1}{(x + 1)^{3}} \\ R_{4} (x) & = \frac{x^{4} - 28 x^{3} + 70 x^{2} - 28 x + 1}{(x + 1)^{4}} \\ R_{n} (x) & = (x + 1)^{- n} \sum_{m = 0}^{n} (- 1)^{m} (\binom{2 n}{2 m}) x^{n - m} \end{aligned}

Partial fraction expansion

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

References

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Chebyshev rational functions

Contents

Properties

Recursion

Differential equations

Orthogonality

Expansion of an arbitrary function

Particular values

Partial fraction expansion

References

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Chebyshev rational functions

Properties

Recursion

Differential equations

Orthogonality

Expansion of an arbitrary function

Particular values

Partial fraction expansion

References

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