JacobiPolynomials

For the Jacobi polynomials:

Initial conditions:

$$P_0(x) = 1$$ $$P_1(x) = \frac{1}{2}(\alpha + \beta + 2) + (\alpha - \beta) x$$

Recurrence relation:

$$P_n(x) = a_n x P_{n-1}(x) + b_n P_{n-2}(x)$$

Common factor:

$$c_n = \frac{2n + \alpha + \beta - 1}{(2n + \alpha + \beta)(2n - 2 + \alpha + \beta)}$$

Simplified coefficients:

$$a_n = c_n (\alpha^2 - \beta^2)$$ $$b_n = c_n (2n + \alpha + \beta)$$