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Chebyshev rational functions
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<h2 id="properties">Properties</h2> <p>Many properties can be derived from the properties of the Chebyshev polynomials of the first kind. Other properties are unique to the functions themselves. </p> <h3>Recursion</h3> 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} <h3>Differential equations</h3> ( 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} <h3>Orthogonality</h3> <p>Defining: </p> ω ( x ) = d e f 1 ( x + 1 ) x {\displaystyle \omega (x)\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{(x+1){\sqrt {x}}}}} <p>The orthogonality of the Chebyshev rational functions may be written: </p> ∫ 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}} <p>where <i>cn</i> = 2 for <i>n</i> = 0 and <i>cn</i> = 1 for <i>n</i> ≥ 1; <i>δnm</i> is the <a href="/facts/Kronecker_delta/gcGAy0bm">Kronecker delta</a> function. </p> <h3>Expansion of an arbitrary function</h3> <p>For an arbitrary function <i>f</i>(<i>x</i>) ∈ <i>L</i>2<i>ω</i> the orthogonality relationship can be used to expand <i>f</i>(<i>x</i>): </p> f ( x ) = ∑ n = 0 ∞ F n R n ( x ) {\displaystyle f(x)=\sum _{n=0}^{\infty }F_{n}R_{n}(x)} <p>where </p> 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.} <h2 id="particular-values">Particular values</h2> 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}}} <h2 id="partial-fraction-expansion">Partial fraction expansion</h2> 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}}}} <ul><li>Guo, Ben-Yu; Shen, Jie; Wang, Zhong-Qing (2002). <a href="http://www.math.purdue.edu/~shen/pub/GSW_IJNME02.pdf">"Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval"</a> (PDF). <i>Int. J. Numer. Methods Eng</i>. 53 (1): 65–84. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2002IJNME..53...65G">2002IJNME..53...65G</a>. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.121.6069">10.1.1.121.6069</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1002%2Fnme.392">10.1002/nme.392</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:9208244">9208244</a>. Retrieved 2006-07-25.</li></ul>