EQL (version 1.0-1)

hermite: Hermite Polynomials

Description

Computes the Hermite polynomial \(H_n(x)\).

Usage

hermite(x, n, prob = TRUE)

Arguments

x

a numeric vector or array giving the values at which the Hermite polynomial should be evaluated.

n

an integer vector or array giving the degrees of the Hermite polynomials. If length(x) != 1, n must be either of the same length as x or a single value.

prob

logical. If TRUE (the default) the probabilistic version of the Hermite polynomial is evaluated, otherwise the physicists' Hermite polynomials are used. See the ‘Details’ section below for further information.

Value

the Hermite polynomial (either the probabilists' or the physicists' version) evaluated at x.

Details

The Hermite polynomials are given by:

• \(H_{n+1}(x)=xH_n(x)-nH_{n-1}(x),\) with \(H_0(x)=1\) and \(H_1(x)=x\), (Probabilists' version \(H_n^{Pr}(x)\))

• \(H_{n+1}(x)=2xH_n(x)-2nH_{n-1}(x)\), with \(H_0(x)=1\) and \(H_1(x)=2x\). (Physicists' version \(H_n^{Ph}(x)\))

and the relationship between the two versions is given by $$H_n^{Ph}(x)=2^{n/2}H_n^{Pr}(\sqrt{2}x). $$ The term ‘probabilistic’ is motivated by the fact that in this case the Hermite polynomial \(H_n(x)\) can be as well defined by $$H_n(x)=(-1)^{n}\frac{1}{\varphi(x)} \varphi^{(n)}(x),$$ where \(\varphi(x)\) denotes the density function of the standard normal distribution and \(\varphi^{(k)}(x)\) denotes the \(k\)th derivative of \(\varphi(x)\) with respect to \(x\).

If the argument n is a vector it must be of the same length as the argument x or the length of the argument x must be equal to one. The Hermite polynomials are then evaluated either at \(x_i\) with degree \(n_i\) or at \(x\) with degree \(n_i\), respectively.

References

Fedoryuk, M.V. (2001). Hermite polynomials. Encyclopaedia of Mathematics, Kluwer Academic Publishers.

Examples

# NOT RUN {
2^(3/2)*hermite(sqrt(2)*5, 3)    # = 940
hermite(5, 3, FALSE)             # = 940
hermite(2:4, 1:3)                # H_1(2), H_2(3), H_3(4)
hermite(2:4, 2)                  # H_2(2), H_2(3), H_2(4)
hermite(2, 1:3)                  # H_1(2), H_2(2), H_3(2)
# }
# NOT RUN {
hermite(1:3, 1:4)                # Error!
# }