Jacobipolynom

Med hjälp av hypergeometriska funktionen

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

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

Rodrigues formel

En alternativ definition ges av Rodirgues formel

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

Ortogonalitet

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.

Symmetrirelation

Jacobipolynomen satisfierar symmetrirelationen

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

Derivator

Jacobipolynomens kte derivata ges av

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

Differentialekvation

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

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

Differensekvation

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, ....

Generenade funktion

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

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