jacobi(n, alpha, beta, monic=False)
Defined to be the solution of
$$(1 - x^2)\frac{d^2}{dx^2}P_n^{(\alpha, \beta)} + (\beta - \alpha - (\alpha + \beta + 2)x) \frac{d}{dx}P_n^{(\alpha, \beta)} + n(n + \alpha + \beta + 1)P_n^{(\alpha, \beta)} = 0$$for $\alpha, \beta > -1$ ; $P_n^{(\alpha, \beta)}$ is a polynomial of degree $n$ .
For fixed $\alpha, \beta$ , the polynomials $P_n^{(\alpha, \beta)}$ are orthogonal over $[-1, 1]$ with weight function $(1 - x)^\alpha(1 + x)^\beta$ .
Degree of the polynomial.
Parameter, must be greater than -1.
Parameter, must be greater than -1.
If :None:None:`True`, scale the leading coefficient to be 1. Default is :None:None:`False`.
Jacobi polynomial.
Jacobi polynomial.
The Jacobi polynomials satisfy the recurrence relation:
$$P_n^{(\alpha, \beta-1)}(x) - P_n^{(\alpha-1, \beta)}(x) = P_{n-1}^{(\alpha, \beta)}(x)$$This can be verified, for example, for $\alpha = \beta = 2$ and $n = 1$ over the interval $[-1, 1]$ :
>>> import numpy as np
... from scipy.special import jacobi
... x = np.arange(-1.0, 1.0, 0.01)
... np.allclose(jacobi(0, 2, 2)(x),
... jacobi(1, 2, 1)(x) - jacobi(1, 1, 2)(x)) True
Plot of the Jacobi polynomial $P_5^{(\alpha, -0.5)}$ for different values of $\alpha$ :
>>> import matplotlib.pyplot as plt
... import numpy as np
... from scipy.special import jacobi
... x = np.arange(-1.0, 1.0, 0.01)
... fig, ax = plt.subplots()
... ax.set_ylim(-2.0, 2.0)
... ax.set_title(r'Jacobi polynomials $P_5^{(\alpha, -0.5)}$')
... for alpha in np.arange(0, 4, 1):
... ax.plot(x, jacobi(5, alpha, -0.5)(x), label=rf'$\alpha={alpha}$')
... plt.legend(loc='best')
... plt.show()
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.special._orthogonal.jacobi
scipy.special._orthogonal.chebyt
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