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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:

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 $(1 - x^2)^{(\alpha - 1/2)}$.