Inom matematiken är Jacobipolynomen en viktig klass ortogonala polynom. De introducerades av Carl Gustav Jacob Jacobi. Flera andra ortogonala polynom är specialfall av dem, däribland Gegenbauerpolynomen, Legendrepolynomen, Zernikepolynomen samt Tjebysjovpolynomen.
Jacobipolynomen kan definieras via hypergeometriska funktionen enligt
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bed04644d7795bb63b3b0fd7e14081f79810672)
där
är Pochhammersymbolen. Ett ekvivalent uttyck är
![{\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}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb72ba6ff8911f3a9abcf06af7a42148ddd33dfb)
En alternativ definition ges av Rodirgues formel
![{\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\}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9b7f7a8a445d5bc2f2608b99b849847ad8cfa1)
![{\displaystyle P_{0}^{(\alpha ,\beta )}(z)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab1be0010d74fdc47d7429f18dbbf8063925f6c)
![{\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)
Jacobipolynomen satisfierar ortogonalitetsrelationen
![{\displaystyle \int _{-1}^{1}(1-x)^{\alpha }(1+x)^{\beta }P_{m}^{(\alpha ,\beta )}(x)P_{n}^{(\alpha ,\beta )}(x)\;dx={\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20ed5359673c6e83a25ca9c03e463701ef4774c3)
för α, β > −1.
Jacobipolynomen satisfierar symmetrirelationen
![{\displaystyle P_{n}^{(\alpha ,\beta )}(-z)=(-1)^{n}P_{n}^{(\beta ,\alpha )}(z);}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2600d9ed56e859d5a202e9e439a6cbf4512bfd0a)
Jacobipolynomens kte derivata ges av
![{\displaystyle {\frac {\mathrm {d} ^{k}}{\mathrm {d} z^{k}}}P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (\alpha +\beta +n+1+k)}{2^{k}\Gamma (\alpha +\beta +n+1)}}P_{n-k}^{(\alpha +k,\beta +k)}(z).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa63a834344746d8b863181ede59fd9a4bf4115f)
Jacobipolynomet Pn(α, β) är en lösning av andra ordningens linjära homogena differentialekvation
![{\displaystyle (1-x^{2})y''+(\beta -\alpha -(\alpha +\beta +2)x)y'+n(n+\alpha +\beta +1)y=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/333f8464a161577178cb7f2cedecbc06c2ba5b52)
Jacobipolynomen satisfierar differensekvationen
![{\displaystyle {\begin{aligned}&2n(n+\alpha +\beta )(2n+\alpha +\beta -2)P_{n}^{(\alpha ,\beta )}(z)=\\&\quad =(2n+\alpha +\beta -1){\Big \{}(2n+\alpha +\beta )(2n+\alpha +\beta -2)z+\alpha ^{2}-\beta ^{2}{\Big \}}P_{n-1}^{(\alpha ,\beta )}(z)-2(n+\alpha -1)(n+\beta -1)(2n+\alpha +\beta )P_{n-2}^{(\alpha ,\beta )}(z)~,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3d0316797f2fa2d3be946874db43652dae023d)
för n = 2, 3, ....
Jacobipolynomens genererande funktion ges av
![{\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 }~}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e53f2c37d917968933494795622b253df5dc382)
där
![{\displaystyle R=R(z,w)=\left(1-2zw+w^{2}\right)^{\frac {1}{2}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a6d58927b46350ae56cee4fd6d29a951b185542)
![{\displaystyle P_{n}^{(\alpha ,\beta )}(1)={n+\alpha \choose n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c46e240dcb9a1b7bc98ebeb1d4d80355a7c10d)
![{\displaystyle P_{n}^{(\alpha ,\beta )}(-1)=(-1)^{n}{n+\beta \choose n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b31e46dfad32476cf22779390a80a70bc44b0bbd)
Jacobipolynomen satisfierar
![{\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/9fb4579ac22055f383d83e2bf6d4b24e7f215aab)
En annan formel är
![{\displaystyle P_{n}^{(\alpha ,\beta )}(\cos \theta )={\frac {\cos \left(\left[n+(\alpha +\beta +1)/2\right]\theta -\left[2\alpha +1\right]\pi /4\right)}{{\sqrt {\pi n}}\left[\sin(\theta /2)\right]^{\alpha +1/2}\left[\cos(\theta /2)\right]^{\beta +1/2}}}+{\mathcal {O}}\left(n^{-3/2}\right),~~~0<\theta <\pi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfbdc9975955211fd6a45f99cf645ae262f42a68)
- Den här artikeln är helt eller delvis baserad på material från engelskspråkiga Wikipedia, Jacobi polynomials, 4 december 2013.
Speciella funktioner |
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| Gamma- och relaterade funktioner | | | Zeta- och L-funktioner | | | Besselfunktioner och relaterade funktioner | | | Elliptiska funktioner och thetafunktioner | | | Hypergeometriska funktioner | | | Ortogonala polynom | | | Andra funktioner | |
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