scipy.special.gegenbauer

scipy.special.gegenbauer(n, alpha, monic=False)

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

nint

Degree of the polynomial.

monicbool, optional

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

Returns

Corthopoly1d

Gegenbauer polynomial.

Notes

The polynomials \(C_n^{(\alpha)}\) are orthogonal over \([-1,1]\) with weight function \((1 - x^2)^{(\alpha - 1/2)}\).