`scipy.special._basic:assoc_laguerre`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_basic.py:1572

Signature

def assoc_laguerre(x, n, k = 0.0)

Summary

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

Extended Summary

This function evaluates the generalized Laguerre polynomial

L_{n}^{(k)} (x) .

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

e^{- x} x^{k}

for $k > - 1$ .

Parameters

x : float or ndarray: Points where to evaluate the Laguerre polynomial
n : int: Degree of the Laguerre polynomial
k : int: 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

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

import numpy as np
from scipy.special import assoc_laguerre, eval_genlaguerre

✓

np.isclose(assoc_laguerre(1, 3, 2), 7/3)

✗

np.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 :math:`L_3^{(k)}` for several values of :math:`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")

✗

<matplotlib.legend.Legend object at 0x7f3f60e61a90>

plt.show()

✓

Aliases

scipy.special.assoc_laguerre