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scipy.special.

# chebys

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

  Chebyshev polynomial of the second kind on \([-2, 2]\).

  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 \([-2, 2]\).

  See also

  `chebyu`

    Chebyshev polynomial of the second kind

  Notes

  The polynomials \(S_n(x)\) are orthogonal over \([-2, 2]\) with weight function \(\sqrt{1 - (x/2)^2}\).

  References

  [1]

  Abramowitz and Stegun, “Handbook of Mathematical Functions” Section 22. National Bureau of Standards, 1972.

  Examples

  Evaluate the Chebyshev polynomial of the second kind \(S_3\) at \(x = 1\):

      >>> import numpy as np
      >>> from scipy.special import chebys, chebyu
      >>> np.isclose(chebys(3)(1), -1.0)
      True

  The polynomial \(S_n\) 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 \(S_n\) for several values of \(n\):

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