scipy.special.eval_sh_jacobi#

scipy.special.eval_sh_jacobi(n, p, q, x, out=None) = <ufunc 'eval_sh_jacobi'>#

Evaluate shifted Jacobi polynomial at a point.

Defined by

\[G_n^{(p, q)}(x) = \frac{\Gamma(n+1)\Gamma(n+p)}{\Gamma(2n+p)} P_n^{(p - q, q - 1)}(2x - 1),\]

where \(P_n^{(\cdot, \cdot)}\) is the n-th Jacobi polynomial. See 22.5.2 in [AS] (or equivalently [DLMF]) for details. Note that here, in contrast to eval_sh_legendre, eval_sh_chebyt, and eval_sh_chebyu, more than just the argument is shifted.

Parameters:

nfloat: Degree of the polynomial. If not an integer, the result is determined via the relation to binom and eval_jacobi.
pfloat: Parameter, must satisfy \(p - q > -1\).
qfloat: Parameter, must be greater than 0.
xarray_like: Points at which to evaluate the polynomial.
outndarray, optional: Optional output array for the function values

Returns:

Gscalar or ndarray: Values of the shifted Jacobi polynomial.

See also

roots_sh_jacobi: roots and quadrature weights of shifted Jacobi polynomials
sh_jacobi: shifted Jacobi polynomial object
eval_jacobi: evaluate Jacobi polynomials

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/18.1.E2

Examples

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

These are the parameter values that we’ll use in this example.

>>> p = 0.5
>>> q = 0.125

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

>>> eval_sh_jacobi(3, p, q, [0, 0.25, 0.5])
array([-0.00344968,  0.01217532, -0.02333604])

Plot the polynomials of order 0 through 4.

>>> x = np.linspace(0, 1, 800)
>>> fig, ax = plt.subplots(1, 1, constrained_layout=True)
>>> for n in range(5):
...     y = eval_sh_jacobi(n, p, q, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Shifted Jacobi Polynomials")
>>> ax.set_xlabel("$x$")
>>> ax.set_ylabel("$G_n^{(p, q)}(x)$")
>>> ax.legend(loc='upper left', bbox_to_anchor=(1, 1), shadow=True)
>>> plt.show()

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