Euler matrices and their algebraic properties revisited

This paper is concerned with the generalized Euler polynomial matrix ${\text{E}}^{(\alpha )}(x)$ and the Euler matrix $\text{E}$. Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for ${\text{E}}^{(\alpha )}(x)$ and determine the inverse matrix of $\text{E}$. We establish some explicit expressions for the Euler polynomial matrix $\text{E}(x)$, which involving the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in terms of Stirling matrices, as well as, a connection between the shifted Euler matrices and Vandermonde matrices.