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.

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.

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.

Interaction of 5-acetyl-8-quinolinol, corresponding 1-ethylquinolinium iodides, and 8-hydroxyquinoline-7-carbaldehyde with diethylmalonate under basic condition gave the corresponding of 4–6. Further reaction of 4–6 with hydrazine, phenyl hydrazine, hydroxylamine, urea, and/or thiourea gave the corresponding 1-[8-hydroxy-5(7)-quinolinyl]alkylidene-substituted heterocyclic derivatives (7–16) and corresponding 1-ethylquinolinium iodides (17–21). The structure of the synthesized compounds was confirmed by elemental and spectral analysis. The biological activity of 5-substituted 8-quinolinol was tested as microbicidal and bioregulator agents, the results obtained were interpreted according to their structure-activity relationships.