JacobiPolynomials

Jacobi Polynomials

$$ \begin{aligned} C(n) &= 2n + \alpha + \beta \\ F(n) &= C(n-1) \cdot C(n) \\ G &= \alpha^2 - \beta^2 \\ A(n, x) &= \left(F(n) \cdot x + G\right)\left(\frac{C(n) - 1}{2}\right) \\ B(n) &= (n + \alpha - 1)(n + \beta - 1)C(n) \\ E(n) &= n \cdot C\left(\frac{n}{2}\right) \cdot C(n - 1) \\ P(0,x) &= 1 \\ P(1,x) &= \frac{C(1) \cdot x + \alpha - \beta}{2} \\ P(n, x) &= \frac{A(n, x) \cdot P(n - 1, x) - B(n) \cdot P(n - 2, x)}{E(n)} \forall n \ge 2 \end{aligned} $$