JacobiPolynomials

The Jacobi Polynomials

$$P_n(x) = \left\{ \begin{array}{ll} n = 0 & 1 \\\ n = 1 & \frac{\alpha - \beta + x(2 + \alpha + \beta)}{2} \\\ n = 2 & A_n x P_{n - 1}(x) - B_n P_{n - 2}(x) \end{array} \right.$$

Where:

$$\begin{array}{ll} A_n & = \frac{(2n + \alpha + \beta - 1)(a^2 - b^2 + (2n + \alpha + \beta - 2)(2n + \alpha + \beta))}{2n(n + \alpha + \beta)(2n + \alpha + \beta - 2)} \\\ & = \frac{(C_n - 1)(\alpha^2 - \beta^2 + C_{n - 1} C_n)}{2nC_{\frac{n}{2}} C_{n - 1}} \\\ & \\\ B_n & = \frac{(n + \alpha - 1)(n + \beta - 1)(2n + \alpha + \beta)}{n(n + \alpha + \beta)(2n + \alpha + \beta - 2)} \\\ & = \frac{(n + \alpha - 1)(n + \beta - 1) C_n}{nC_{\frac{n}{2}} C_{n - 1}} \\\ & \\\ C_n & = 2n + \alpha + \beta \end{array}$$