scipy.special.genlaguerre#
- scipy.special.genlaguerre(n, alpha, monic=False)[source]#
Generalized (associated) Laguerre polynomial.
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 \(n\).
- Parameters:
- nint
Degree of the polynomial.
- alphafloat
Parameter, must be greater than -1.
- monicbool, optional
If True, scale the leading coefficient to be 1. Default is False.
- Returns:
- Lorthopoly1d
Generalized Laguerre polynomial.
Notes
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 \(\alpha = 0\).
References
[AS]Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.
Examples
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 \([-1, 1]\):
>>> import numpy as np >>> 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 >>> 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()
