`scipy.special._orthogonal:sh_chebyu`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:3269

Signature

def sh_chebyu(n, monic = False)

Summary

Shifted Chebyshev polynomial of the second kind.

Extended Summary

Defined as $U_{n}^{*} (x) = U_{n} (2 x - 1)$ for $U_{n}$ 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

U : orthopoly1d: Shifted Chebyshev polynomial of the second kind.

Notes

The polynomials $U_{n}^{*}$ are orthogonal over $[0, 1]$ with weight function $(x - x^{2})^{1/2}$ .

Examples

Evaluate the shifted Chebyshev polynomial of the second kind :math:`U^*_3` at :math:`x = 0.75`:

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

✓

np.isclose(sh_chebyu(3)(0.75), -1.0)

✗

np.True_

The polynomial :math:`U^*_n` is the Chebyshev polynomial :math:`U_n` shifted from :math:`[-1, 1]` to :math:`[0, 1]`:

x = np.linspace(0, 1, 5)
np.allclose(sh_chebyu(3)(x), chebyu(3)(2*x - 1))

✓

True

Plot :math:`U^*_n` for several values of :math:`n`:

import matplotlib.pyplot as plt
x = np.linspace(0, 1, 400)
fig, ax = plt.subplots()

✓

for n in range(4):
    ax.plot(x, sh_chebyu(n)(x), label=rf"$U^*_{n}$")
ax.set_title(r"Shifted Chebyshev polynomials $U^*_n$")
ax.set_xlabel("x")
ax.legend(loc="best")

✗

<matplotlib.legend.Legend object at 0x7f3f590a2cf0>

plt.show()

✓

Aliases

scipy.special.sh_chebyu