# scipy.special.laguerre

scipy.special.laguerre(n, monic=False)

  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:
    n int

      Degree of the polynomial.

    monic bool, optional

      If True, scale the leading coefficient to be 1. Default is False.

  Returns:
    L orthopoly1d

      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()