scipy.special.laguerre#

scipy.special.laguerre(n, monic=False)[source]#

Laguerre polynomial.

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 \(n\).

Parameters:

nint: Degree of the polynomial.
monicbool, optional: If True, scale the leading coefficient to be 1. Default is False.

Returns:

Lorthopoly1d: Laguerre Polynomial.

See also

genlaguerre: Generalized (associated) Laguerre polynomial.

Notes

The polynomials \(L_n\) are orthogonal over \([0, \infty)\) with weight function \(e^{-x}\).

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 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 \([-1, 1]\):

>>> import numpy as np
>>> 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 \(n = 3\):

>>> 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
>>> 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()

../../_images/scipy-special-laguerre-1.png