`scipy.special._orthogonal:chebys`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:2901

Signature

def chebys(n, monic = False)

Summary

Chebyshev polynomial of the second kind on $[- 2, 2]$ .

Extended Summary

Defined as $S_{n} (x) = U_{n} (x /2)$ where $U_{n}$ is the nth Chebyshev polynomial of the second kind.

Parameters

n : int: Degree of the polynomial.
monic : bool, optional: If True, scale the leading coefficient to be 1. Default is False.

Returns

S : orthopoly1d: Chebyshev polynomial of the second kind on $[- 2, 2]$ .

Notes

The polynomials $S_{n} (x)$ are orthogonal over $[- 2, 2]$ with weight function $1 - (x /2)^{2}$ .

Examples

Evaluate the Chebyshev polynomial of the second kind :math:`S_3` at :math:`x = 1`:

import numpy as np
from scipy.special import chebys, chebyu

✓

np.isclose(chebys(3)(1), -1.0)

✗

np.True_

The polynomial :math:`S_n` is a scaled Chebyshev polynomial of the second kind:

x = np.linspace(-2, 2, 5)
np.allclose(chebys(3)(x), chebyu(3)(x/2))

✓

True

Plot :math:`S_n` for several values of :math:`n`:

import matplotlib.pyplot as plt
x = np.linspace(-2, 2, 400)
fig, ax = plt.subplots()

✓

for n in range(4):
    ax.plot(x, chebys(n)(x), label=rf"$S_{n}$")
ax.set_title(r"Chebyshev polynomials $S_n$")
ax.set_xlabel("x")
ax.legend(loc="best")

✗

<matplotlib.legend.Legend object at 0x7f3f687d9160>

plt.show()

✓

Aliases

scipy.special.chebys