Some properties of general-$\lambda$-matrix polynomials: an umbral approach

Abstract

The "2-variable general-$\lambda$-matrix polynomials (2VG$\lambda$MP)" is a new family of matrix polynomials, introduced and studied in this article. These matrix polynomials are constructed using umbral and symbolic methods. We delve into the generating function, explicit series representation, differential equation, quasi-monomiality, summation formula, determinant representation, and various identities satisfied by these polynomials. Furthermore, a thorough investigation is conducted to ascertain the outcomes for the members of the general-$\lambda$-matrix family. Additionally, 3D graphs and figures showing the zeros distribution, real zeros, and stacks of zeros for a few members of the family of these polynomials are also presented in the article.