scipy.special.eval_gegenbauer

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

Evaluate Gegenbauer polynomial at a point.

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

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

When \(n\) is an integer the result is a polynomial of degree \(n\). See 22.5.46 in [AS] 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.