In this paper, we introduce a new generalization of the Hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems. An explicit representation and an expansion of the matrix exponential in a series of these matrix polynomials are obtained. Properties of Hermite matrix polynomials such as the recurrence formula permit an efficient computations of matrix functions are established. A new expansion of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.

In this paper the definition of Hermite-Hermite matrix polynomials is introduced starting from the Hermite matrix polynomials. An explicit representation, a matrix recurrence relation for the Hermite-Hermite matrix polynomials is given and differential equations satisfied by them are presented. A new expansion of the matrix exponential for a wide class of matrices in terms of Hermite-Hermite matrix polynomials are proposed.

The oxidative status and the morphological changes of liver of rats exposed to cadmium (5 mg Cd/kg body weight subcutaneously) for 22 days and the protective role of melatonin (10 mg/kg b.w.) against the toxicity of cadmium was studied. The concentration of malondialdehyde (MDA), as an indicator of lipid peroxidation, activity of the antioxidant enzyme superoxide dismutase (SOD) as well as the concentration of glutathione (GSH) was measured in the liver. The morphological changes were investigated using both light and electron microscopes. The exposure to Cd led to an increase of MDA levels and a decrease of both the activity of SOD and GSH concentration in the liver. In contrast, melatonin administration restored the previous changes to nearly the normal levels. Morphologically, Cd led to different histopathological changes such as loss of normal architecture of the parenchymatous tissue, cytoplasmic vacuolization, cellular degeneration and necrosis, congested blood vessels, destructed cristae mitochondria, fat globules, severe glycogen depletion, lipofuscin pigments, and collagenous fibers formation. Again, melatonin administration counteracts all changes and the tissue appears more or less normal. The rate of recovery was faster when melatonin was administered for treatment after the exposure to cadmium than if the animals left without any treatment. The results suggest that melatonin may be useful as an antioxidant in combating free radical-induced oxidative stress and tissue injury that is a result of cadmium toxicity.