polhermite

Function: polhermite
Section: polynomials
C-Name: polhermite_eval0
Prototype: LDGD0,L,
Help: polhermite(n,{a='x},{flag=0}): Hermite polynomial H(n,v) of degree n,
 evaluated at a. If flag is nonzero, return [H_{n-1}(a), H_n(a)].
Description:
  (small,?var):gen    polhermite($1,$2)
  (small,gen):gen     polhermite_eval($1,$2)
Doc: $n^{\text{th}}$ \idx{Hermite} polynomial $H_n$ evaluated at $a$
 (\kbd{'x} by default), i.e.
 $$ H_n(x) = (-1)^n\*e^{x^2} \dfrac{d^n}{dx^n}e^{-x^2}.$$
 If \fl\ is nonzero and $n > 0$, return $[H_{n-1}(a), H_n(a)]$.
 \bprog
 ? polhermite(5)
 %1 = 32*x^5 - 160*x^3 + 120*x
 ? polhermite(5, -2) \\ H_5(-2)
 %2 = 16
 ? polhermite(5,,1)
 %3 = [16*x^4 - 48*x^2 + 12, 32*x^5 - 160*x^3 + 120*x]
 ? polhermite(5,-2,1)
 %4 = [76, 16]
 @eprog

Variant: The variant \fun{GEN}{polhermite}{long n, long v} returns the $n$-th
 Hermite polynomial in variable $v$. To obtain $H_n(a)$,
 use \fun{GEN}{polhermite_eval}{long n, GEN a}.