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 \(\alpha = 0\).