JacobiPolynomials

Jacobi Polynomials

Sequence Definition:

Define a sequence $d_n$ as: $$d_n = 2n + \alpha + \beta$$

Initial conditions:

• $P_0(x) = 1$
• $P_1(x) = \frac{d_1}{2} + (\alpha - \beta) x$

Recurrence relation:

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

Common factor:

$$c_n = \frac{d_n - 1}{d_n \cdot d_{n-1}}$$

Simplified coefficients:

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

$$b_n = c_n \cdot d_n$$