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Jacobi

Jacobi polynomial of order nn is denoted by Pnα,β(x)P_{n}^{\alpha,\beta}\left(x\right). Here α\alpha and β\beta are parameters of Jacobi polynomial. Note that

1+α>0,1+β>01+\alpha>0,\quad1+\beta>0

The weight for Jacobi polynomial is given by

w(x)=(1−x)α(1+x)βw(x)=(1-x)^{\alpha}(1+x)^{\beta}

The support of Jacobi polynomial is [−1,1].[-1,1].

The leading coefficient of Pnα,βP_{n}^{\alpha,\beta} is denoted by knk_{n} and it is given by

kn=12n(2n+α+βn)k_{n}=\frac{1}{2^{n}}\left(\begin{array}{c} 2n+\alpha+\beta\\ n \end{array}\right)

The norm of Pnα,βP_{n}^{\alpha,\beta} is given below:

∥Pnα,β∥2=:hn=2α+β+12n+α+β+1Γ(n+α+1)Γ(n+β+1)n!Γ(n+α+β+1)\Vert P_{n}^{\alpha,\beta}\Vert^{2}=:h_{n}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{n!\Gamma(n+\alpha+\beta+1)}

The value of Pnα,βP_{n}^{\alpha,\beta} at x=1x=1 is given by

Pnα,β(1)=Γ(n+α+1)n!Γ(1+α)P_{n}^{\alpha,\beta}(1)=\frac{\Gamma(n+\alpha+1)}{n!\Gamma(1+\alpha)}

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