  We investigate the global density of zeros of generalized Hermite orthogonal
polynomials, subject to certain truncated conditions on its weight. We shall
given explicitly the global density of zeros under some asymptotic conditions
on the weight. Moreover we compute the asymptotic of the total energy of the
equilibrium position of the system of $n$ movable unit charges in an external
field determined by the weight of the generalized Hermite polynomials. We will
see that for finite $n$ the energy is in direct relationship with the zeros of
the orthogonal polynomials.
