scipy.special.

assoc_laguerre

scipy.special.assoc_laguerre(x, n, k=0.0)

Compute the generalized (associated) Laguerre polynomial of degree \(n\) and order \(k\).

This function evaluates the generalized Laguerre polynomial

\[L^{(k)}_n(x).\]

The polynomial is orthogonal over \([0, \infty)\) with weight function

\[e^{-x}x^k\]

for \(k > -1\).

Parameters:

xfloat or ndarray

Points where to evaluate the Laguerre polynomial

nint

Degree of the Laguerre polynomial

kint

Order of the Laguerre polynomial

Returns:

assoc_laguerre: float or ndarray

Associated laguerre polynomial values

Notes

assoc_laguerre is a simple wrapper around eval_genlaguerre, with reversed argument order (x, n, k=0.0) --> (n, k, x).

Examples

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Evaluate the associated Laguerre polynomial \(L_3^{(2)}\) at \(x = 1\):

>>> import numpy as np
>>> from scipy.special import assoc_laguerre, eval_genlaguerre
>>> np.isclose(assoc_laguerre(1, 3, 2), 7/3)
True

assoc_laguerre is equivalent to eval_genlaguerre with reversed argument order:

>>> x = np.linspace(0, 5, 6)
>>> np.allclose(assoc_laguerre(x, 3, 2), eval_genlaguerre(3, 2, x))
True

Plot \(L_3^{(k)}\) for several values of \(k\):

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 8, 400)
>>> fig, ax = plt.subplots()
>>> for k in range(3):
...     ax.plot(x, assoc_laguerre(x, 3, k), label=rf"$k={k}$")
>>> ax.set_title(r"Associated Laguerre polynomials $L_3^{(k)}$")
>>> ax.set_xlabel("x")
>>> ax.legend(loc="best")
>>> plt.show()

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