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# scipy.special.gegenbauer

scipy.special.gegenbauer(n, alpha, monic=False)

  Gegenbauer (ultraspherical) polynomial.

  Defined to be the solution of

  \[(1 - x^2)\frac{d^2}{dx^2}C_n^{(\alpha)} - (2\alpha + 1)x\frac{d}{dx}C_n^{(\alpha)} + n(n + 2\alpha)C_n^{(\alpha)} = 0\]

  for \(\alpha > -1/2\); \(C_n^{(\alpha)}\) is a polynomial of degree \(n\).

  Parameters:
    n int

      Degree of the polynomial.

    alpha float

      Parameter, must be greater than -0.5.

    monic bool, optional

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

  Returns:
    C orthopoly1d

      Gegenbauer polynomial.

  Notes

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

  Examples

      >>> import numpy as np
      >>> from scipy import special
      >>> import matplotlib.pyplot as plt

  We can initialize a variable `p` as a Gegenbauer polynomial using the `gegenbauer` function and evaluate at a point `x = 1`.

      >>> p = special.gegenbauer(3, 0.5, monic=False)
      >>> p
      poly1d([ 2.5,  0. , -1.5,  0. ])
      >>> p(1)
      1.0

  To evaluate `p` at various points `x` in the interval `(-3, 3)`, simply pass an array `x` to `p` as follows:

      >>> x = np.linspace(-3, 3, 400)
      >>> y = p(x)

  We can then visualize `x, y` using `matplotlib.pyplot`.

      >>> fig, ax = plt.subplots()
      >>> ax.plot(x, y)
      >>> ax.set_title("Gegenbauer (ultraspherical) polynomial of degree 3")
      >>> ax.set_xlabel("x")
      >>> ax.set_ylabel("G_3(x)")
      >>> plt.show()