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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.

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 $\alpha = 0$.