JacobiPolynomials

Let $P_n^{(\alpha, \beta)} (x)=P_n (x)$ throughout for the sake of limiting the proliferation of repetitive notation. The traditional definition of Jacobi polynomials uses the following recurrence relation:

$$ P_n^{(\alpha, \beta)} (x) = (A_{n - 1} x + B_{n - 1}) P_{n - 1}^{(\alpha, \beta)} (x) - C_{n - 1} P_{n - 2}^{(\alpha, \beta)} (x) $$

where

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

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

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

A more concise alternative definition of the Jacobi polynomials that is better suited for computation is given by

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

where

$$ a_n = \frac{b_n}{c_n} (\alpha^2 - \beta^2) $$

$$ b_n = \frac{c_n - 1}{c_{n - 1}} $$

$$ c_n = 2 n + \alpha + \beta $$

It can be shown that these two recurrence relations are equivalent by substituting the expressions for (a_n, b_n, c_n) into the alternative definition and simplifying. Starting with the alternative definition:

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

Substituting the expressions for $a_n, b_n, c_n$

$$ P_n (x) = \left( \frac{b_n}{c_n} (\alpha^2 - \beta^2) \right) xP_{n - 1} (x) + \left( \frac{c_n - 1}{c_{n - 1}} \right) P_{n - 2} (x) $$

$$ P_n (x) = \left( \frac{(c_n - 1)}{c_{n - 1}} (\alpha^2 - \beta^2) \right) xP_{n - 1} (x) + \left( \frac{c_n - 1}{c_{n - 1}} \right) P_{n - 2} (x) $$

Factoring out $(c_n - 1) / c_{n - 1}$

$$ P_n (x) = \frac{(c_n - 1)}{c_{n - 1}} ((\alpha^2 - \beta^2) xP_{n - 1} (x) + P_{n - 2} (x)) $$

Substituting $c_n = 2 n + \alpha + \beta$

$$ P_n (x) = \frac{(2 n + \alpha + \beta - 1)}{2 n + \alpha + \beta - 1} ((\alpha^2 - \beta^2) xP_{n - 1} (x) + P_{n - 2} (x)) $$

Cancelling common factors:

$$ P_n (x) = (A_{n - 1} x + B_{n - 1}) P_{n - 1} (x) - C_{n - 1} P_{n - 2} (x) $$

Therefore, the alternative definition of Jacobi polynomials using the parameters $c_n, a_n, b_n$ is equivalent to the standard definition in terms of $A_n, B_n, C_n$.

Specific Instances

ChebyshevPolynomial