hermite_polynomial_he

→Tensor

hermite_polynomial_he(x: Tensor, n: int)

source

Probabilist's Hermite polynomial $\mathit{He}_n(x)$ .

Probabilists' Hermite polynomials are orthogonal on $(-\infty, \infty)$ with weight $e^{-x^2/2} / \sqrt{2\pi}$ — the standard normal density — and are the natural basis for Hermite-expansions of functions against a Gaussian measure (Wiener chaos, Gauss-Hermite quadrature with weight 1).

Parameters

xTensor

Argument tensor; any floating-point dtype.

nint

Non-negative polynomial degree.

Returns

Tensor

$\mathit{He}_n(x)$ element-wise, same shape and dtype as x.

Notes

Recurrence:

\mathit{He}_0(x) = 1, \quad \mathit{He}_1(x) = x, \quad \mathit{He}_{k+1}(x) = x\, \mathit{He}_k(x) - k\, \mathit{He}_{k-1}(x).

Connection to physicists' Hermite: $\mathit{He}_n(x) = 2^{-n/2} H_n(x/\sqrt 2)$ . Raises ValueError for n < 0.

Examples

>>> import lucid
>>> from lucid.special import hermite_polynomial_he
>>> hermite_polynomial_he(lucid.tensor([-1.0, 0.0, 1.0]), n=3)
Tensor([2.0000, 0.0000, -2.0000])