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 \([-1, 1]\):

>>> 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
>>> from scipy.special import genlaguerre
>>> 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()