Description Usage Arguments Details Value Author(s) References Examples
Computes the Hermite polynomial H_n(x).
1 |
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 |
prob |
logical. If |
The Hermite polynomials are given by:
H_(n+1)(x) = x*H_n(x) - n*H_(n-1)(x), with H_0(x)=1 and H_1(x)=x, (Probabilists' version H_n^Pr(x))
H_(n+1)(x) = 2*x*H_n(x) - 2*n*H_(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(2^(1/2)*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*phi^(n)(x)/phi(x),
where phi(x) denotes the density function of the standard normal distribution and phi^(k)(x) denotes the kth derivative of phi(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.
the Hermite polynomial (either the probabilists' or the
physicists' version) evaluated at x
.
Thorn Thaler
Fedoryuk, M.V. (2001). Hermite polynomials. Encyclopaedia of Mathematics, Kluwer Academic Publishers.
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