  We study systems of holomorphic Hermite functions in the Segal-Bargmann
spaces, which are Hilbert spaces of entire functions on the complex Euclidean
space, and are determined by the Bargmann-type integral transform on the real
Euclidean space. We prove that for any positive parameter which is strictly
smaller than the minimum eigenvalue of the positive Hermitian matrix associated
with the transform, one can find a generator of holomorphic Hermite functions
whose anihilation and creation operators satisfy canonical commutation
relations. In other words, we find the necessary and sufficient conditions so
that some kinds of entire functions can be such generators. Moreover, we also
study the complete orthogonality, the eigenvalue problems and the Rodrigues
formulas.
