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scipy.special.

# genlaguerre

scipy.special.genlaguerre(n, alpha, monic=False)

  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:
    n int

      Degree of the polynomial.

    alpha float

      Parameter, must be greater than -1.

    monic bool, optional

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

  Returns:
    L orthopoly1d

      Generalized Laguerre polynomial.

  See also

  `laguerre`

    Laguerre polynomial.

  `hyp1f1`

    confluent hypergeometric function

  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()