Bernoulli matrix and its algebraic properties

Abstract

In this paper, we define the generalized Bernoulli polynomial matrix B ( α ) ( x ) and the Bernoulli matrix B . Using some properties of Bernoulli polynomials and numbers, a product formula of B ( α ) ( x ) and the inverse of B were given. It is shown that not only B ( x ) = P [ x ] B , where P [ x ] is the generalized Pascal matrix, but also B ( x ) = FM ( x ) = N ( x ) F , where F is the Fibonacci matrix, M ( x ) and N ( x ) are the ( n + 1 ) × ( n + 1 ) lower triangular matrices whose ( i , j ) -entries are i j B i - j ( x ) - i - 1 j B i - j - 1 ( x ) - i - 2 j B i - j - 2 ( x ) and i j B i - j ( x ) - i j + 1 B i - j - 1 ( x ) - i j + 2 B i - 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.