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:

nint

Degree of the polynomial.

alphafloat

Parameter, must be greater than -1.

monicbool, optional

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

Returns:

Lorthopoly1d

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()