scipy.special.eval_gegenbauer#

scipy.special.eval_gegenbauer(n, alpha, x, out=None) = <ufunc 'eval_gegenbauer'>#

Evaluate Gegenbauer (ultraspherical) polynomial at a point.

The Gegenbauer polynomials can be defined via the Gauss hypergeometric function \({}_2F_1\) as

\[C_n^{(\alpha)}(x) = \frac{(2\alpha)_n}{\Gamma(n + 1)} {}_2F_1(-n, 2\alpha + n; \alpha + 1/2; (1 - x)/2).\]

where \((\cdot)_n\) is the Pochhammer symbol; see poch. When \(n\) is an integer the result is a polynomial of degree \(n\). See 22.5.46 in [AS] (or equivalently [DLMF]) for details.

Parameters:

narray_like: Degree of the polynomial. If not an integer, the result is determined via the relation to the Gauss hypergeometric function.
alphaarray_like: Parameter.
xarray_like: Points at which to evaluate the Gegenbauer polynomial.
outndarray, optional: Optional output array for the function values.

Returns:

Cscalar or ndarray: Values of the Gegenbauer polynomial.

See also

roots_gegenbauer: roots and quadrature weights of Gegenbauer polynomials
gegenbauer: Gegenbauer polynomial object
hyp2f1: Gauss hypergeometric function

References

[AS]

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

[DLMF]

NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/18.5.E9

Examples

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

This is the parameter value that we’ll use in this example.

>>> alpha = 1.25

Evaluate the order 3 polynomial at a few values of x.

>>> eval_gegenbauer(3, alpha, [0, 0.25, 0.5])
array([ 0.        , -1.21582031, -1.2890625 ])

Plot the polynomials of order 0 through 4.

>>> x = np.linspace(-1, 1, 800)
>>> fig, ax = plt.subplots(1, 1, constrained_layout=True)
>>> for n in range(5):
...     y = eval_gegenbauer(n, alpha, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Gegenbauer Polynomials")
>>> ax.set_xlabel("$x$")
>>> ax.set_ylabel(r"$C_n^{(\alpha)}(x)$")
>>> ax.legend(loc='upper left', bbox_to_anchor=(1, 1), shadow=True)
>>> plt.show()

../../_images/scipy-special-eval_gegenbauer-1.png