code: def test_hermite_prob():
      assert hermite_prob(0, x) == 1
      assert hermite_prob(1, x) == x
      assert hermite_prob(2, x) == x**2 - 1
      assert hermite_prob(3, x) == x**3 - 3 * x
      assert hermite_prob(4, x) == x**4 - 6 * x**2 + 3
      assert (
          hermite_prob(6, x) == x**6 - 15 * x**4 + 45 * x**2 - 15
      )
  
      n = Symbol("n")
      assert unchanged(hermite_prob, n, x)
      assert hermite_prob(n, -x) == (-1) ** n * hermite_prob(n, x)
      assert unchanged(hermite_prob, -n, x)
  
      assert hermite_prob(n, 0) == sqrt(pi) / gamma(S.Half - n / 2)
      assert hermite_prob(n, oo) is oo
  
      assert conjugate(hermite_prob(n, x)) == hermite_prob(
          n, conjugate(x)
      )
  
      _k = Dummy("k")
      assert (
          hermite_prob(n, x)
          .rewrite(Sum)
          .dummy_eq(
              factorial(n)
              * Sum(
                  (-S.Half) ** _k
                  * x ** (n - 2 * _k)
                  / (factorial(_k) * factorial(n - 2 * _k)),
                  (_k, 0, floor(n / 2)),
              )
          )
      )
      assert (
          hermite_prob(n, x)
          .rewrite("polynomial")
          .dummy_eq(
              factorial(n)
              * Sum(
                  (-S.Half) ** _k
                  * x ** (n - 2 * _k)
                  / (factorial(_k) * factorial(n - 2 * _k)),
                  (_k, 0, floor(n / 2)),
              )
          )
      )
  
      assert diff(hermite_prob(n, x), x) == n * hermite_prob(n - 1, x)
      assert diff(hermite_prob(n, x), n) == Derivative(
          hermite_prob(n, x), n
      )
      raises(ArgumentIndexError, lambda: hermite_prob(n, x).fdiff(3))
  
      assert hermite_prob(n, x).rewrite(hermite) == sqrt(2) ** (
          -n
      ) * hermite(n, x / sqrt(2))
  
imports: from sympy.concrete.summations import Sum
  from sympy.core.function import Derivative, diff
  from sympy.core.numbers import Rational, oo, pi, zoo
  from sympy.core.singleton import S
  from sympy.core.symbol import Dummy, Symbol
  from sympy.functions.combinatorial.factorials import RisingFactorial, binomial, factorial
  from sympy.functions.elementary.complexes import conjugate
  from sympy.functions.elementary.exponential import exp
  from sympy.functions.elementary.integers import floor
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.functions.elementary.trigonometric import cos
  from sympy.functions.special.gamma_functions import gamma
  from sympy.functions.special.hyper import hyper
  from sympy.functions.special.polynomials import assoc_laguerre, assoc_legendre, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, gegenbauer, hermite, hermite_prob, jacobi, jacobi_normalized, laguerre, legendre
  from sympy.polys.orthopolys import laguerre_poly
  from sympy.polys.polyroots import roots
  from sympy.core.expr import unchanged
  from sympy.core.function import ArgumentIndexError
  from sympy.testing.pytest import raises
label_desc: Use image node_id 7 for calling a global function with example usage: test_hermite_prob() without return types