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Gegenbauerovi polinomi su ortogonalni polinomi
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{\displaystyle C_{n}^{(\alpha )}}
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{\displaystyle (1-x^{2})y''-(2\alpha +1)xy'+n(n+2\alpha )y=0.\,}
Gegenbauerovi polinomi predstavljaju specijalni slučaj Jakobijevih polinoma , a Ležandrovi polinomi i Čebiševljevi polinomi su specijalni slučaj Gegenbauerovih polinoma. Dobili su ime po austrijskom matematičaru Leopoldu Gegenbaueru .
Gegenbauerovi polinomi su specijalni slučaj Jakobijevih polinoma :
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{\displaystyle C_{n}^{(\alpha )}(x)={\frac {(2\alpha )_{n}}{(\alpha +{\frac {1}{2}})_{n}}}P_{n}^{(\alpha -1/2,\alpha -1/2)}(x).}
Mogu da se prikažu pomoću hipergeometrijske funkcije :
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{\displaystyle C_{n}^{(\alpha )}(z)={\frac {(2\alpha )_{n}}{n!}}\,_{2}F_{1}\left(-n,2\alpha +n;\alpha +{\frac {1}{2}};{\frac {1-z}{2}}\right).}
odnosno razvojem se dobija:
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{\displaystyle C_{n}^{(\alpha )}(z)=\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{k}{\frac {\Gamma (n-k+\alpha )}{\Gamma (\alpha )k!(n-2k)!}}(2z)^{n-2k}.}
Gegenbauerovi polinomi mogu da se prikažu i pomoću Rodrigezove formule:
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{\displaystyle C_{n}^{(\alpha )}(z)=\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{k}{\frac {\Gamma (n-k+\alpha )}{\Gamma (\alpha )k!(n-2k)!}}(2z)^{n-2k}.}
Funkcija generatrisa Gegenbauerovih polinoma je:
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{\displaystyle {\frac {1}{(1-2xt+t^{2})^{\alpha }}}=\sum _{n=0}^{\infty }C_{n}^{(\alpha )}(x)t^{n}.}
Gegenbauerovi polinomi zadovoljavaju sledeću rekurziju:
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{\displaystyle {\begin{aligned}C_{0}^{\alpha }(x)&=1\\C_{1}^{\alpha }(x)&=2\alpha x\\C_{n}^{\alpha }(x)&={\frac {1}{n}}[2x(n+\alpha -1)C_{n-1}^{\alpha }(x)-(n+2\alpha -2)C_{n-2}^{\alpha }(x)].\end{aligned}}}
Za fiksni α polinomi su ortogonalni na [−1, 1] sa težinskom funkcijom:
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{\displaystyle w(z)=\left(1-z^{2}\right)^{\alpha -{\frac {1}{2}}}.}
Dobija se za n ≠ m ,
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{\displaystyle \int _{-1}^{1}C_{n}^{(\alpha )}(x)C_{m}^{(\alpha )}(x)(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx=0.}
a za isti n :
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{\displaystyle \int _{-1}^{1}\left[C_{n}^{(\alpha )}(x)\right]^{2}(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )[\Gamma (\alpha )]^{2}}}.}
![{\displaystyle {\begin{aligned}C_{0}^{\alpha }(x)&=1\\C_{1}^{\alpha }(x)&=2\alpha x\\C_{n}^{\alpha }(x)&={\frac {1}{n}}[2x(n+\alpha -1)C_{n-1}^{\alpha }(x)-(n+2\alpha -2)C_{n-2}^{\alpha }(x)].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f7e040cb401c72d41eab0bda662f115229feed)
![{\displaystyle \int _{-1}^{1}\left[C_{n}^{(\alpha )}(x)\right]^{2}(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )[\Gamma (\alpha )]^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b5696990f291fed55949f3b9e2f1669b9f4c83)