scipy.special.

sh_chebyu

scipy.special.sh_chebyu(n, monic=False)

Shifted Chebyshev polynomial of the second kind.

Defined as \(U^*_n(x) = U_n(2x - 1)\) for \(U_n\) the nth Chebyshev polynomial of the second kind.

Parameters:

nint

Degree of the polynomial.

monicbool, optional

If True, scale the leading coefficient to be 1. Default is False.

Returns:

Uorthopoly1d

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

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Evaluate the shifted Chebyshev polynomial of the second kind \(U^*_3\) at \(x = 0.75\):

>>> import numpy as np
>>> from scipy.special import chebyu, sh_chebyu
>>> np.isclose(sh_chebyu(3)(0.75), -1.0)
True

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

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

Plot \(U^*_n\) for several values of \(n\):

>>> 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")
>>> plt.show()

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