genlaguerre(n, alpha, monic=False)
Defined to be the solution of
$$x\frac{d^2}{dx^2}L_n^{(\alpha)} + (\alpha + 1 - x)\frac{d}{dx}L_n^{(\alpha)} + nL_n^{(\alpha)} = 0,$$where $\alpha > -1$ ; $L_n^{(\alpha)}$ is a polynomial of degree $n$ .
For fixed $\alpha$ , the polynomials $L_n^{(\alpha)}$ are orthogonal over $[0, \infty)$ with weight function $e^{-x}x^\alpha$ .
The Laguerre polynomials are the special case where $\alpha = 0$ .
Degree of the polynomial.
Parameter, must be greater than -1.
If :None:None:`True`, scale the leading coefficient to be 1. Default is :None:None:`False`.
Generalized Laguerre polynomial.
Generalized (associated) Laguerre polynomial.
hyp1f1
confluent hypergeometric function
laguerre
Laguerre polynomial.
The generalized Laguerre polynomials are closely related to the confluent hypergeometric function ${}_1F_1$ :
$$L_n^{(\alpha)} = \binom{n + \alpha}{n} {}_1F_1(-n, \alpha +1, x)$$
This can be verified, for example, for $n = \alpha = 3$ over the interval $[-1, 1]$ :
>>> from scipy.special import binom
... from scipy.special import genlaguerre
... from scipy.special import hyp1f1
... x = np.arange(-1.0, 1.0, 0.01)
... np.allclose(genlaguerre(3, 3)(x), binom(6, 3) * hyp1f1(-3, 4, x)) True
This is the plot of the generalized Laguerre polynomials $L_3^{(\alpha)}$ for some values of $\alpha$ :
>>> import matplotlib.pyplot as plt
... from scipy.special import genlaguerre
... x = np.arange(-4.0, 12.0, 0.01)
... fig, ax = plt.subplots()
... ax.set_ylim(-5.0, 10.0)
... ax.set_title(r'Generalized Laguerre polynomials $L_3^{\alpha}$')
... for alpha in np.arange(0, 5):
... ax.plot(x, genlaguerre(3, alpha)(x), label=rf'$L_3^{(alpha)}$')
... 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.genlaguerre
scipy.special._orthogonal.laguerre
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