Exploring Zeros of Hermite-λ Matrix Polynomials: A Numerical Approach

This article aims to introduce a set of hybrid matrix polynomials associated with λ-polynomials and explore their properties using a symbolic approach. The main outcomes of this study include the derivation of generating functions, series definitions, and differential equations for the newly introduced two-variable Hermite λ-matrix polynomials. Furthermore, we establish the quasi-monomiality property of these polynomials, derive summation formulae and integral representations, and examine the graphical representation and symmetric structure of their approximate zeros using computer-aided programs. Finally, this article concludes by introducing the idea of 1-variable Hermite λ matrix polynomials and their structure of zeros using a computer-aided program.