hermite {EQL} | R Documentation |
Computes the Hermite polynomial H_n(x)
.
hermite(x, n, prob = TRUE)
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)=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.
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.
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!
## End(Not run)