`scipy.special._orthogonal:sh_jacobi`

source: /scipy/special/_orthogonal.py:464

Signature

def sh_jacobi(n, p, q, monic = False)

Summary

Shifted Jacobi polynomial.

Extended Summary

Defined by

G_{n}^{(p, q)} (x) = (n 2 n + p - 1)^{- 1} P_{n}^{(p - q, q - 1)} (2 x - 1),

where $P_{n}^{(\cdot, \cdot)}$ is the nth Jacobi polynomial.

Parameters

n : int: Degree of the polynomial.
p : float: Parameter, must have $p > q - 1$ .
q : float: Parameter, must be greater than 0.
monic : bool, optional: If True, scale the leading coefficient to be 1. Default is False.

Returns

G : orthopoly1d: Shifted Jacobi polynomial.

Notes

For fixed $p, q$ , the polynomials $G_{n}^{(p, q)}$ are orthogonal over $[0, 1]$ with weight function $(1 - x)^{p - q} x^{q - 1}$ .

Examples

Evaluate the shifted Jacobi polynomial :math:`G_3^{(2, 1)}` at :math:`x = 0.5`:

import numpy as np
from scipy.special import binom, jacobi, sh_jacobi

✓

np.isclose(sh_jacobi(3, 2, 1)(0.5), -3/280)

✗

np.True_

The polynomial is related to the Jacobi polynomial :math:`P_n^{(p - q, q - 1)}`:

x = np.linspace(0, 1, 5)
n, p, q = 3, 2, 1
scale = 1 / binom(2*n + p - 1, n)
np.allclose(sh_jacobi(n, p, q)(x),
            scale * jacobi(n, p - q, q - 1)(2*x - 1))

✓

True

Plot :math:`G_3^{(p, 1)}` for several values of :math:`p`:

import matplotlib.pyplot as plt
x = np.linspace(0, 1, 400)
fig, ax = plt.subplots()

✓

for p in [1, 2, 3]:
    ax.plot(x, sh_jacobi(3, p, 1)(x), label=rf"$p={p}$")
ax.set_title(r"Shifted Jacobi polynomials $G_3^{(p, 1)}$")
ax.set_xlabel("x")
ax.legend(loc="best")

✗

<matplotlib.legend.Legend object at 0x7fa8e4f0ae40>

plt.show()

✓

Aliases

scipy.special.sh_jacobi