# hermite

##### Hermite Polynomials

Computes univariate and multivariate Hermite polynomials.

##### Usage

`hermite(order, sigma = 1, var = "x")`

##### Arguments

- order
integer. The order of the Hermite polynomial.

- sigma
the covariance matrix of the Gaussian kernel.

- var
character. The variables of the polynomial.

##### Details

Hermite polynomials are obtained by successive differentiation of the Gaussian kernel $$H_{\nu}(x,\Sigma) = exp \Bigl( \frac{1}{2} x^\dagger \Sigma x \Bigl) (- \partial_x )^\nu exp \Bigl( -\frac{1}{2} x^\dagger \Sigma x \Bigl)$$ where \(\Sigma\) is a d-dimensional square matrix and \(\nu=(\nu_1, ..., \nu_d)\) is the vector representing the order of differentiation for each variable.

##### Value

list of Hermite polynomials with components

- f
the Hermite polynomial.

- order
the order of the Hermite polynomial.

- terms
data.frame containing the variables, coefficients and degrees of each term in the Hermite polynomial.

##### Examples

```
# NOT RUN {
# univariate Hermite polynomials up to order 3
hermite(3)
# univariate Hermite polynomials with variable z
hermite(3, var = 'z')
# multivariate Hermite polynomials up to order 2
hermite(order = 2,
sigma = matrix(c(1,0,0,1), nrow = 2),
var = c('z1', 'z2'))
# }
```

*Documentation reproduced from package calculus, version 0.1.0, License: GPL-3*