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

${\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(\alpha +1)_{n)){n!))\,{}_{2}F_{1}\left(-n,1+\alpha +\beta +n;\alpha +1;{\frac {1-z}{2))\right)}$

där ${\displaystyle (\alpha +1)_{n))$ är Pochhammersymbolen. Ett ekvivalent uttyck är

${\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (\alpha +n+1)}{n!\,\Gamma (\alpha +\beta +n+1)))\sum _{m=0}^{n}{n \choose m}{\frac {\Gamma (\alpha +\beta +n+m+1)}{\Gamma (\alpha +m+1)))\left({\frac {z-1}{2))\right)^{m}~.}$

En alternativ definition ges av Rodirgues formel

${\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(-1)^{n)){2^{n}n!))(1-z)^{-\alpha }(1+z)^{-\beta }{\frac {d^{n)){dz^{n))}\left\{(1-z)^{\alpha }(1+z)^{\beta }(1-z^{2})^{n}\right\}~.}$

${\displaystyle P_{0}^{(\alpha ,\beta )}(z)=1}$

${\displaystyle P_{1}^{(\alpha ,\beta )}(z)={\frac {1}{2))\left[2(\alpha +1)+(\alpha +\beta +2)(z-1)\right]}$

${\displaystyle P_{2}^{(\alpha ,\beta )}(z)={\frac {1}{8))\left[4(\alpha +1)(\alpha +2)+4(\alpha +\beta +3)(\alpha +2)(z-1)+(\alpha +\beta +3)(\alpha +\beta +4)(z-1)^{2}\right]}$

Jacobipolynomen satisfierar ortogonalitetsrelationen

${\displaystyle \int _{-1}^{1}(1-x)^{\alpha }(1+x)^{\beta }P_{m}^{(\alpha ,\beta )}(x)P_{n}^{(\alpha ,\beta )}(x)\;dx={\frac {2^{\alpha +\beta +1)){2n+\alpha +\beta +1)){\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{\Gamma (n+\alpha +\beta +1)n!))\delta _{nm))$

för α, β > −1.

Jacobipolynomen satisfierar symmetrirelationen

${\displaystyle P_{n}^{(\alpha ,\beta )}(-z)=(-1)^{n}P_{n}^{(\beta ,\alpha )}(z);}$

Jacobipolynomens kte derivata ges av

${\displaystyle {\frac {\mathrm {d} ^{k)){\mathrm {d} z^{k))}P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (\alpha +\beta +n+1+k)}{2^{k}\Gamma (\alpha +\beta +n+1)))P_{n-k}^{(\alpha +k,\beta +k)}(z).}$

Jacobipolynomet P_n^{(α, β)} är en lösning av andra ordningens linjära homogena differentialekvation

${\displaystyle (1-x^{2})y''+(\beta -\alpha -(\alpha +\beta +2)x)y'+n(n+\alpha +\beta +1)y=0.}$

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))}$

för n = 2, 3, ....

Jacobipolynomens genererande funktion ges av

${\displaystyle \sum _{n=0}^{\infty }P_{n}^{(\alpha ,\beta )}(z)w^{n}=2^{\alpha +\beta }R^{-1}(1-w+R)^{-\alpha }(1+w+R)^{-\beta }~}$

där

${\displaystyle R=R(z,w)=\left(1-2zw+w^{2}\right)^{\frac {1}{2))~.}$

${\displaystyle P_{n}^{(\alpha ,\beta )}(1)={n+\alpha \choose n))$

${\displaystyle P_{n}^{(\alpha ,\beta )}(-1)=(-1)^{n}{n+\beta \choose n))$

Jacobipolynomen satisfierar

${\displaystyle {\begin{aligned}\lim _{n\to \infty }n^{-\alpha }P_{n}^{(\alpha ,\beta )}\left(\cos {\frac {z}{n))\right)&=\left({\frac {z}{2))\right)^{-\alpha }J_{\alpha }(z)~\\\lim _{n\to \infty }n^{-\beta }P_{n}^{(\alpha ,\beta )}\left(\cos \left[\pi -{\frac {z}{n))\right]\right)&=\left({\frac {z}{2))\right)^{-\beta }J_{\beta }(z)~\end{aligned)).}$

En annan formel är

${\displaystyle P_{n}^{(\alpha ,\beta )}(\cos \theta )={\frac {\cos \left(\left[n+(\alpha +\beta +1)/2\right]\theta -\left[2\alpha +1\right]\pi /4\right)}((\sqrt {\pi n))\left[\sin(\theta /2)\right]^{\alpha +1/2}\left[\cos(\theta /2)\right]^{\beta +1/2))}+{\mathcal {O))\left(n^{-3/2}\right),~~~0<\theta <\pi .}$

• Askey–Gaspers olikhet

Den här artikeln är helt eller delvis baserad på material från engelskspråkiga Wikipedia, Jacobi polynomials, 4 december 2013.