Bernoulli matrix and its algebraic properties

In this paper, we define the generalized Bernoulli polynomial matrix B(α)(x)B(α)(x) and the Bernoulli matrix BB. Using some properties of Bernoulli polynomials and numbers, a product formula of B(α)(x)B(α)(x) and the inverse of BB were given. It is shown that not only B(x)=P[x]BB(x)=P[x]B, where P[x]P[x] is the generalized Pascal matrix, but also B(x)=FM(x)=N(x)FB(x)=FM(x)=N(x)F, where FF is the Fibonacci matrix, M(x)M(x) and N(x)N(x) are the (n+1)×(n+1)(n+1)×(n+1) lower triangular matrices whose (i,j)(i,j)-entries are ijBi-j(x)-i-1jBi-j-1(x)-i-2jBi-j-2(x) and ijBi-j(x)-ij+1Bi-j-1(x)-ij+2Bi-j-2(x), respectively. From these formulas, several interesting identities involving the Fibonacci numbers and the Bernoulli polynomials and numbers are obtained. The relationships are established about Bernoulli, Fibonacci and Vandermonde matrices.