laguerre(n, monic=False)
Defined to be the solution of
$$x\frac{d^2}{dx^2}L_n + (1 - x)\frac{d}{dx}L_n + nL_n = 0;$$$L_n$ is a polynomial of degree $n$ .
The polynomials $L_n$ are orthogonal over $[0, \infty)$ with weight function $e^{-x}$ .
Degree of the polynomial.
If :None:None:`True`, scale the leading coefficient to be 1. Default is :None:None:`False`.
Laguerre Polynomial.
Laguerre polynomial.
genlaguerre
Generalized (associated) Laguerre polynomial.
The Laguerre polynomials $L_n$ are the special case $\alpha = 0$ of the generalized Laguerre polynomials $L_n^{(\alpha)}$ . Let's verify it on the interval $[-1, 1]$ :
>>> from scipy.special import genlaguerre
... from scipy.special import laguerre
... x = np.arange(-1.0, 1.0, 0.01)
... np.allclose(genlaguerre(3, 0)(x), laguerre(3)(x)) True
The polynomials $L_n$ also satisfy the recurrence relation:
$$(n + 1)L_{n+1}(x) = (2n +1 -x)L_n(x) - nL_{n-1}(x)$$This can be easily checked on $[0, 1]$ for $n = 3$ :
>>> from scipy.special import laguerre
... x = np.arange(0.0, 1.0, 0.01)
... np.allclose(4 * laguerre(4)(x),
... (7 - x) * laguerre(3)(x) - 3 * laguerre(2)(x)) True
This is the plot of the first few Laguerre polynomials $L_n$ :
>>> import matplotlib.pyplot as plt
... from scipy.special import laguerre
... x = np.arange(-1.0, 5.0, 0.01)
... fig, ax = plt.subplots()
... ax.set_ylim(-5.0, 5.0)
... ax.set_title(r'Laguerre polynomials $L_n$')
... for n in np.arange(0, 5):
... ax.plot(x, laguerre(n)(x), label=rf'$L_{n}$')
... 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.genlaguerre
scipy.special._orthogonal.laguerre
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