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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 is defined as:

R n ( x )   = d e f   T n ( x − 1 x + 1 ) {\displaystyle R_{n}(x)\ {\stackrel {\mathrm {def} }{=}}\ T_{n}\left({\frac {x-1}{x+1}}\right)}

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

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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 ( x − 1 x + 1 ) R n ( x ) − R n − 1 ( x ) for n ≥ 1 {\displaystyle R_{n+1}(x)=2\left({\frac {x-1}{x+1}}\right)R_{n}(x)-R_{n-1}(x)\quad {\text{for}}\,n\geq 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 ≥ 2 {\displaystyle (x+1)^{2}R_{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\geq 2} ( x + 1 ) 2 x d 2 d x 2 R n ( x ) + ( 3 x + 1 ) ( x + 1 ) 2 d d x R n ( x ) + n 2 R n ( x ) = 0 {\displaystyle (x+1)^{2}x{\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^{2}R_{n}(x)=0}

Orthogonality

Defining:

ω ( x )   = d e f   1 ( x + 1 ) x {\displaystyle \omega (x)\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{(x+1){\sqrt {x}}}}}

The orthogonality of the Chebyshev rational functions may be written:

∫ 0 ∞ R m ( x ) R n ( x ) ω ( x ) d x = π c n 2 δ n m {\displaystyle \int _{0}^{\infty }R_{m}(x)\,R_{n}(x)\,\omega (x)\,\mathrm {d} x={\frac {\pi c_{n}}{2}}\delta _{nm}}

where cn = 2 for n = 0 and cn = 1 for n ≥ 1; δnm is the Kronecker delta function.

Expansion of an arbitrary function

For an arbitrary function f(x) ∈ L2ω the orthogonality relationship can be used to expand f(x):

f ( x ) = ∑ n = 0 ∞ F n R n ( x ) {\displaystyle f(x)=\sum _{n=0}^{\infty }F_{n}R_{n}(x)}

where

F n = 2 c n π ∫ 0 ∞ f ( x ) R n ( x ) ω ( x ) d x . {\displaystyle F_{n}={\frac {2}{c_{n}\pi }}\int _{0}^{\infty }f(x)R_{n}(x)\omega (x)\,\mathrm {d} x.}

Particular values

R 0 ( x ) = 1 R 1 ( x ) = x − 1 x + 1 R 2 ( x ) = x 2 − 6 x + 1 ( x + 1 ) 2 R 3 ( x ) = x 3 − 15 x 2 + 15 x − 1 ( x + 1 ) 3 R 4 ( x ) = x 4 − 28 x 3 + 70 x 2 − 28 x + 1 ( x + 1 ) 4 R n ( x ) = ( x + 1 ) − n ∑ m = 0 n ( − 1 ) m ( 2 n 2 m ) x n − m {\displaystyle {\begin{aligned}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{aligned}}}

Partial fraction expansion

R n ( x ) = ∑ m = 0 n ( m ! ) 2 ( 2 m ) ! ( n + m − 1 m ) ( n m ) ( − 4 ) m ( x + 1 ) m {\displaystyle 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}}}}