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

TL;DR

This paper introduces and analyzes a new family of matrix polynomials called 2-variable general-$\lambda$-matrix polynomials, using umbral methods, exploring their properties, identities, and zeros distribution.

Contribution

It presents the definition, properties, and identities of the new 2VG$\lambda$MP family, including generating functions, differential equations, and zeros analysis, using umbral and symbolic techniques.

Findings

Derived explicit series and determinant representations.

Established differential equations and identities for the polynomials.

Analyzed zeros distribution through graphical representations.

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.