scipy.special.gegenbauer¶
- scipy.special.gegenbauer(n, alpha, monic=False)[source]¶
Gegenbauer (ultraspherical) polynomial.
Defined to be the solution of
\[(1 - x^2)\frac{d^2}{dx^2}C_n^{(\alpha)} - (2\alpha + 1)x\frac{d}{dx}C_n^{(\alpha)} + n(n + 2\alpha)C_n^{(\alpha)} = 0\]for \(\alpha > -1/2\); \(C_n^{(\alpha)}\) is a polynomial of degree \(n\).
Parameters: n : int
Degree of the polynomial.
monic : bool, optional
If True, scale the leading coefficient to be 1. Default is False.
Returns: C : orthopoly1d
Gegenbauer polynomial.
Notes
The polynomials \(C_n^{(\alpha)}\) are orthogonal over \([-1,1]\) with weight function \((1 - x^2)^{(\alpha - 1/2)}\).