In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)}
are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight
( 1 − x ) α ( 1 + x ) β {\displaystyle (1-x)^{\alpha }(1+x)^{\beta }} on the interval [ − 1 , 1 ] {\displaystyle [-1,1]} . The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.
Definitions
Via the hypergeometric function The Jacobi polynomials are defined via the hypergeometric function as follows:
P n ( α , β ) ( z ) = ( α + 1 ) n n !
2 F 1 ( − n , 1 + α + β + n ; α + 1 ; 1 2 ( 1 − z ) ) , {\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(\alpha +1)_{n}}{n!}}\,{}_{2}F_{1}\left(-n,1+\alpha +\beta +n;\alpha +1;{\tfrac {1}{2}}(1-z)\right),}
where ( α + 1 ) n {\displaystyle (\alpha +1)_{n}} is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometric function is finite, therefore one obtains the following equivalent expression:
P n ( α , β ) ( z ) = Γ ( α + n + 1 ) n ! Γ ( α + β + n + 1 ) ∑ m = 0 n ( n m ) Γ ( α + β + n + m + 1 ) Γ ( α + m + 1 ) ( z − 1 2 ) m . {\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' formula An equivalent definition is given by Rodrigues' formula:
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