`scipy.special._orthogonal:sh_chebyt`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:3090

Signature

def sh_chebyt(n, monic = False)

Summary

Shifted Chebyshev polynomial of the first kind.

Extended Summary

Defined as $T_{n}^{*} (x) = T_{n} (2 x - 1)$ for $T_{n}$ the nth Chebyshev polynomial of the first kind.

Parameters

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

Returns

T : orthopoly1d: Shifted Chebyshev polynomial of the first kind.

Notes

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

Examples

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

import numpy as np
from scipy.special import chebyt, sh_chebyt

✓

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

✗

np.True_

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

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

✓

True

Plot :math:`T^*_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_chebyt(n)(x), label=rf"$T^*_{n}$")
ax.set_title(r"Shifted Chebyshev polynomials $T^*_n$")
ax.set_xlabel("x")
ax.legend(loc="best")

✗

<matplotlib.legend.Legend object at 0x7f3f5917fa10>

plt.show()

✓

Aliases

scipy.special.sh_chebyt