laguerre_polynomial_l

→Tensor

laguerre_polynomial_l(x: Tensor, n: int)

source

Laguerre polynomial $L_n(x)$ .

The (simple) Laguerre polynomials are orthogonal on $[0, \infty)$ with weight $e^{-x}$ and appear as the radial eigenfunctions of the hydrogen atom, as the basis of Gauss-Laguerre quadrature, and in survival-analysis kernels.

Parameters

xTensor

Argument tensor; any floating-point dtype. Customarily evaluated on

[0, \infty)

nint

Non-negative polynomial degree.

Returns

Tensor

$L_n(x)$ element-wise, same shape and dtype as x.

Notes

Recurrence:

L_0(x) = 1, \quad L_1(x) = 1 - x, \quad (k+1)\, L_{k+1}(x) = (2k + 1 - x)\, L_k(x) - k\, L_{k-1}(x).

The leading coefficient is $(-x)^n / n!$ , and $L_n(0) = 1$ for every n. Raises ValueError for n < 0.

Examples

>>> import lucid
>>> from lucid.special import laguerre_polynomial_l
>>> laguerre_polynomial_l(lucid.tensor([0.0, 1.0, 2.0]), n=2)
Tensor([1.0000, -0.5000, -1.0000])