hermite_polynomial_h

→Tensor

hermite_polynomial_h(x: Tensor, n: int)

source

Physicist's Hermite polynomial $H_n(x)$ .

The physicists' Hermite polynomials are orthogonal on $(-\infty, \infty)$ with weight $e^{-x^2}$ and form the eigenfunctions of the quantum harmonic oscillator (in conjunction with the Gaussian envelope).

Parameters

xTensor

Argument tensor; any floating-point dtype.

nint

Non-negative polynomial degree.

Returns

Tensor

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

Notes

Rodrigues formula and recurrence:

H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}, \qquad H_{k+1}(x) = 2x\, H_k(x) - 2k\, H_{k-1}(x).

Starting values: $H_0(x) = 1$ , $H_1(x) = 2x$ . Linked to the probabilists' Hermite polynomials by $H_n(x) = 2^{n/2} He_n(\sqrt{2}\, x)$ . Raises ValueError for n < 0.

Examples

>>> import lucid
>>> from lucid.special import hermite_polynomial_h
>>> hermite_polynomial_h(lucid.tensor([-1.0, 0.0, 1.0]), n=3)
Tensor([4.0000, 0.0000, -4.0000])