scipy.special.eval_jacobi

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

Evaluate Jacobi polynomial at a point.

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

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

where \((\cdot)_n\) is the Pochhammer symbol; see poch. When \(n\) is an integer the result is a polynomial of degree \(n\). See 22.5.42 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

betaarray_like

Parameter

xarray_like

Points at which to evaluate the polynomial

outndarray, optional

Optional output array for the function values

Returns:

Pscalar or ndarray

Values of the Jacobi polynomial

See also

roots_jacobi

roots and quadrature weights of Jacobi polynomials

jacobi

Jacobi 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.