scipy.special.jacobi

scipy.special.jacobi(n, alpha, beta, monic=False)

Jacobi polynomial.

Defined to be the solution of

\[(1 - x^2)\frac{d^2}{dx^2}P_n^{(\alpha, \beta)} + (\beta - \alpha - (\alpha + \beta + 2)x) \frac{d}{dx}P_n^{(\alpha, \beta)} + n(n + \alpha + \beta + 1)P_n^{(\alpha, \beta)} = 0\]

for \(\alpha, \beta > -1\); \(P_n^{(\alpha, \beta)}\) is a polynomial of degree \(n\).

Parameters:

n : int

Degree of the polynomial.

alpha : float

Parameter, must be greater than -1.

beta : float

Parameter, must be greater than -1.

monic : bool, optional

If True, scale the leading coefficient to be 1. Default is False.

Returns:

P : orthopoly1d

Jacobi polynomial.

Notes

For fixed \(\alpha, \beta\), the polynomials \(P_n^{(\alpha, \beta)}\) are orthogonal over \([-1, 1]\) with weight function \((1 - x)^\alpha(1 + x)^\beta\).