`scipy.special._orthogonal:gegenbauer`

Signature

def gegenbauer(n, alpha, monic = False)

Gegenbauer (ultraspherical) polynomial.

Defined to be the solution of the second-order linear ordinary differential equation

(1 - x^{2}) \frac{d ^{2}}{d x ^{2}} C_{n}^{(α)} (x) - (2 α + 1) x \frac{d}{d x} C_{n}^{(α)} (x) + n (n + 2 α) C_{n}^{(α)} (x) = 0

for $α > - 1/2$ ; $C_{n}^{(α)}$ is a polynomial of degree $n$ .

n : int: Degree of the polynomial.
alpha : float: Parameter, must be greater than -0.5.
monic : bool, optional: If True, scale the leading coefficient to be 1. Default is False.

The polynomials $C_{n}^{(α)}$ are orthogonal on $[- 1, 1]$ with respect to the weight function $(1 - x^{2})^{(α - 1/2)}$ .

import numpy as np
from scipy import special
import matplotlib.pyplot as plt

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We can initialize a variable ``p`` as a Gegenbauer polynomial using the `gegenbauer` function and evaluate at a point ``x = 1``.

p = special.gegenbauer(3, 0.5, monic=False)
p

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poly1d([ 2.5,  0. , -1.5,  0. ])

p(1)

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np.float64(1.0)

To evaluate ``p`` at various points ``x`` in the interval ``(-3, 3)``, simply pass an array ``x`` to ``p`` as follows:

x = np.linspace(-3, 3, 400)
y = p(x)

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We can then visualize ``x, y`` using `matplotlib.pyplot`.

fig, ax = plt.subplots()

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ax.plot(x, y)
ax.set_title("Gegenbauer (ultraspherical) polynomial of degree 3")
ax.set_xlabel("x")
ax.set_ylabel(r"$C_3^{(0.5)}(x)$")

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Text(0, 0.5, '$C_3^{(0.5)}(x)$')

plt.show()

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