Chebyshev rational functions explained

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 is defined as:

R _{n(x) \stackrel{def} }\ T_n\left(\frac\right)

where 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( (x -1)/(x +1) \right)R _{n} (x) - R _{n-1} (x) for n\ge 1

Differential equations

(x +1 ) ^{2 R} _{n(x) = (1)/(n +1)} (d)/(d x) R _{n+1} (x) - (1)/(n -1) (d)/(d x) R _{n-1} (x) for n\ge 2 (x +1 ) ^{2 x (d ^{2})/(d x ^{2})} R _{n(x) + (( 3 x +1 )(x +1 ))/(2)} (d)/(d x) R ^{2 R} _{n} (x) = 0

Orthogonality

Defining:

\omega(x) \stackrel{ def }\ \frac

The orthogonality of the Chebyshev rational functions may be written:

\int ^{inf ty} _{0} R _{m(x)R} ^{n(x)\omega(x) d x = (\pi c _{n})/(2)} \delta _{nm}

where for and for ; is the Kronecker delta function.

Expansion of an arbitrary function

For an arbitrary function the orthogonality relationship can be used to expand :

f(x) = \sum ^{inf ty} _{n =0} F _{n} R _{n(x)}

where

F _{n = (2)/(c _{n\pi})} \int ^{inf ty} _{0} f(x)R _{n(x)\omega(x)dx.}

Particular values

\begin{align} R _{0(x)&=1\\ R} ^{1 (x)& = (x -1)/(x +1)} \\ R _{2 (x)& = (x ^{2-6x+1})/((x +1 ) ^{2})} \\ R _{3 (x)& = (x ^{3-15x} ^{2+15x-1})/((x +1 ) ^{3})} \\ R _{4 (x)& = (x ^{4-28x} ^{3+70x} ^{2-28x+1})/((x +1 ) ^{4})} \\ R ^{- n} _{n(x)& = (x +1 )} \sum ^{n} _{m =0} ( -1 ) ^{m\binom{2n}{2m}x} ^{n-m} \end{align}

Partial fraction expansion

R _{n(x)=\sum} ^{n} ^{m =0} ((m!) ^{2})/(\binom{n + m -1 }{m}\binom{n}{m} ( 2 m)!) (( -4 ) ^{m})/((x +1 ) ^{m})

References

• Ben-Yu . Guo . Jie . Shen . Zhong-Qing . Wang . 2002 . Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval . Int. J. Numer. Methods Eng. . 53 . 1 . 65–84 . 10.1002/nme.392 . 2002IJNME..53...65G . 2006-07-25 . 10.1.1.121.6069 . 9208244 .