Matrix Representation for the Beta Type Polynomials

The aim of this paper is to study and investigate some new properties of the beta polynomials. Taking derivative of the generating functions for beta type polynomials, we give two partial differential equations (PDEs). By using these PDEs, we derive derivative formulas of the beta type polynomials. In order to construct a matrix representation for the beta polynomials, we firstly show that the set of beta polynomials is linearly independent. By using linearly independent properties, we prove that any polynomial of degree less than and equal n are written as a linearly combination of the beta polynomials. Therefore, we define matrix representation for the beta polynomials. Moreover, we provide the simulation of the beta polynomials with some their graphs. We also give remarks and examples and comments on the beta polynomials and their matrix representation.