Jakobijevi polinomi , često zvani i hipergeometrijski polinomi su klasični ortogonalni polinom predstavljeni formulom:
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{\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (\alpha +n+1)}{n!\,\Gamma (\alpha +\beta +n+1)}}\sum _{m=0}^{n}{n \choose m}{\frac {\Gamma (\alpha +\beta +n+m+1)}{\Gamma (\alpha +m+1)}}\left({\frac {z-1}{2}}\right)^{m}~.}
Gegenbauerovi polinomi , Ležandrovi polinomi i Čebiševljevi polinomi predstavljaju specijalni slučaj Jakobijevih polinoma. Jakobijeve polinome otkrio je 1859 . nemački matematičar Karl Gustav Jakobi .
Diferencijalna jednačina
uredi
Jakobijevi polinomi predstavljaju rešenje linerane homogene diferencijalne jednačine drugoga reda:
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{\displaystyle (1-x^{2})y''+(\beta -\alpha -(\alpha +\beta +2)x)y'+n(n+\alpha +\beta +1)y=0.\,}
Definicija
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Jakobijevi polinomi definisani su pomoću hipergeometrijske funkcije :
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{\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(\alpha +1)_{n}}{n!}}\,_{2}F_{1}\left(-n,1+\alpha +\beta +n;\alpha +1;{\frac {1-z}{2}}\right),}
gde
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{\displaystyle (\alpha +1)_{n}}
Pohhamerov simbol . U tom slučaju razvojem se dobija:
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{\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (\alpha +n+1)}{n!\,\Gamma (\alpha +\beta +n+1)}}\sum _{m=0}^{n}{n \choose m}{\frac {\Gamma (\alpha +\beta +n+m+1)}{\Gamma (\alpha +m+1)}}\left({\frac {z-1}{2}}\right)^{m}~.}
Rodrigezova formula
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Jakobijevi polinomi mogu da se definišu i pomoću Rodrigezove formule:
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{\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(-1)^{n}}{2^{n}n!}}(1-z)^{-\alpha }(1+z)^{-\beta }{\frac {d^{n}}{dz^{n}}}\left\{(1-z)^{\alpha }(1+z)^{\beta }(1-z^{2})^{n}\right\}~.}
Generirajuća funkcija
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Generirajuća funkcija Jakobijevih polinoma je:
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{\displaystyle \sum _{n=0}^{\infty }P_{n}^{(\alpha ,\beta )}(z)w^{n}=2^{\alpha +\beta }R^{-1}(1-w+R)^{-\alpha }(1+w+R)^{-\beta }~,}
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{\displaystyle R=R(z,w)={\big (}1-2zw+w^{2}{\big )}^{1/2}~,}
Relacije rekurzije za Jakobijeve polinome su:
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{\displaystyle {\begin{aligned}&2n(n+\alpha +\beta )(2n+\alpha +\beta -2)P_{n}^{(\alpha ,\beta )}(z)\\&\qquad =(2n+\alpha +\beta -1){\Big \{}(2n+\alpha +\beta )(2n+\alpha +\beta -2)z+\alpha ^{2}-\beta ^{2}{\Big \}}P_{n-1}^{(\alpha ,\beta )}(z)\\&\qquad \qquad -2(n+\alpha -1)(n+\beta -1)(2n+\alpha +\beta )P_{n-2}^{(\alpha ,\beta )}(z)~,\quad n=2,3,\dots \end{aligned}}}
Nekoliko prvih polinoma je:
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{\displaystyle P_{0}^{(\alpha ,\beta )}(z)=1}
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{\displaystyle P_{1}^{(\alpha ,\beta )}(z)={\frac {1}{2}}\left[2(\alpha +1)+(\alpha +\beta +2)(z-1)\right]}
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{\displaystyle P_{2}^{(\alpha ,\beta )}(z)={\frac {1}{8}}\left[4(\alpha +1)(\alpha +2)+4(\alpha +\beta +3)(\alpha +2)(z-1)+(\alpha +\beta +3)(\alpha +\beta +4)(z-1)^{2}\right]}
Izraz za realni argument
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Za realno x Jakobijevi polinomi mogu da se pišu i kao:
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{\displaystyle P_{n}^{(\alpha ,\beta )}(x)=\sum _{s}{n+\alpha \choose s}{n+\beta \choose n-s}\left({\frac {x-1}{2}}\right)^{n-s}\left({\frac {x+1}{2}}\right)^{s}}
gde su s ≥ 0 i n -s ≥ 0, a za celobrojno n
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{\displaystyle {z \choose n}={\frac {\Gamma (z+1)}{\Gamma (n+1)\Gamma (z-n+1)}},}
U gornjoj jednačini Γ (z ) je gama funkcija .
U specijalnom slučaju, kada su n , n +α , n +β , and
n +α +β nenegativni celi brojevi Jakobijevi polinomi mogu da se napišu kao:
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{\displaystyle {\begin{aligned}&P_{n}^{(\alpha ,\beta )}(x)=(n+\alpha )!(n+\beta )!\\&\qquad \times \sum _{s}\left[s!(n+\alpha -s)!(\beta +s)!(n-s)!\right]^{-1}\left({\frac {x-1}{2}}\right)^{n-s}\left({\frac {x+1}{2}}\right)^{s}.\end{aligned}}}
Ortogonalnost
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Jakobijevi polinomi za α > -1 i β > -1 zadovoljavaju uslov ortogonalnosti:
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{\displaystyle {\begin{aligned}&\int _{-1}^{1}(1-x)^{\alpha }(1+x)^{\beta }P_{m}^{(\alpha ,\beta )}(x)P_{n}^{(\alpha ,\beta )}(x)\;dx\\&\quad ={\frac {2^{\alpha +\beta +1}}{2n+\alpha +\beta +1}}{\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{\Gamma (n+\alpha +\beta +1)n!}}\delta _{nm}\end{aligned}}}
Težinska funkcija je bila:
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{\displaystyle (1-x)^{\alpha }(1+x)^{\beta }}
Oni nisu ortonormalni, a za normalizaciju:
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{\displaystyle P_{n}^{(\alpha ,\beta )}(1)={n+\alpha \choose n}.}
Jakobijevi polinomi zadovoljavaju sledeće relacije simetrije:
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{\displaystyle P_{n}^{(\alpha ,\beta )}(-z)=(-1)^{n}P_{n}^{(\beta ,\alpha )}(z);}
pa je
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{\displaystyle P_{n}^{(\alpha ,\beta )}(-1)=(-1)^{n}{n+\beta \choose n}.}
Asimptotski izrazi
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Za x unutar intervala [-1, 1], asimptotska vrednost P n (α ,β ) za veliki n dan je:
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{\displaystyle P_{n}^{(\alpha ,\beta )}(\cos \theta )=n^{-1/2}\cos(N\theta +\gamma )+O(n^{-3/2})~,}
gde
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{\displaystyle {\begin{aligned}k(\theta )&=\pi ^{-1/2}\sin ^{-\alpha -1/2}{\frac {\theta }{2}}\cos ^{-\beta -1/2}{\frac {\theta }{2}}~,\\N&=n+{\frac {\alpha +\beta +1}{2}}~,\\\gamma &=-(\alpha +{\frac {1}{2}}){\frac {\pi }{2}}~,\end{aligned}}}
Asimptote blizu ±1 dane su sa:
lim
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{\displaystyle {\begin{aligned}\lim _{n\to \infty }n^{-\alpha }P_{n}^{\alpha ,\beta }\left(\cos {\frac {z}{n}}\right)&=\left({\frac {z}{2}}\right)^{-\alpha }J_{\alpha }(z)~,\\\lim _{n\to \infty }n^{-\beta }P_{n}^{\alpha ,\beta }\left(\cos \left[\pi -{\frac {z}{n}}\right]\right)&=\left({\frac {z}{2}}\right)^{-\beta }J_{\beta }(z)~,\end{aligned}}}
Veza sa Vignerovom d-matricom
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Jakobijevi polinomi povezani su sa Vignerovom D-matricom :
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{\displaystyle d_{m'm}^{j}(\phi )=\left[{\frac {(j+m)!(j-m)!}{(j+m')!(j-m')!}}\right]^{1/2}\left(\sin {\frac {\phi }{2}}\right)^{m-m'}\left(\cos {\frac {\phi }{2}}\right)^{m+m'}P_{j-m}^{(m-m',m+m')}(\cos \phi ).}
Literatura
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![{\displaystyle P_{1}^{(\alpha ,\beta )}(z)={\frac {1}{2}}\left[2(\alpha +1)+(\alpha +\beta +2)(z-1)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a95e6198603c0eb110abde34809dcd787796025)
![{\displaystyle P_{2}^{(\alpha ,\beta )}(z)={\frac {1}{8}}\left[4(\alpha +1)(\alpha +2)+4(\alpha +\beta +3)(\alpha +2)(z-1)+(\alpha +\beta +3)(\alpha +\beta +4)(z-1)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44cf56154dd63ffe05a063e6733b2e8100cad972)
![{\displaystyle {\begin{aligned}&P_{n}^{(\alpha ,\beta )}(x)=(n+\alpha )!(n+\beta )!\\&\qquad \times \sum _{s}\left[s!(n+\alpha -s)!(\beta +s)!(n-s)!\right]^{-1}\left({\frac {x-1}{2}}\right)^{n-s}\left({\frac {x+1}{2}}\right)^{s}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf8b0987f766a9d3fbfeedf257f60747fc98421)
![{\displaystyle {\begin{aligned}\lim _{n\to \infty }n^{-\alpha }P_{n}^{\alpha ,\beta }\left(\cos {\frac {z}{n}}\right)&=\left({\frac {z}{2}}\right)^{-\alpha }J_{\alpha }(z)~,\\\lim _{n\to \infty }n^{-\beta }P_{n}^{\alpha ,\beta }\left(\cos \left[\pi -{\frac {z}{n}}\right]\right)&=\left({\frac {z}{2}}\right)^{-\beta }J_{\beta }(z)~,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a65b8c5c0053ddaacf22938424f200a6d4a8224e)
![{\displaystyle d_{m'm}^{j}(\phi )=\left[{\frac {(j+m)!(j-m)!}{(j+m')!(j-m')!}}\right]^{1/2}\left(\sin {\frac {\phi }{2}}\right)^{m-m'}\left(\cos {\frac {\phi }{2}}\right)^{m+m'}P_{j-m}^{(m-m',m+m')}(\cos \phi ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2abcaf41d9baf29c60160c839c3d9e062c80ce9e)