genlaguerre#

scipy.special.genlaguerre(n, alpha, monic=False)[source]#

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

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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()

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