Some combinatorial identities via Fibonacci numbers
Discrete Applied Mathematics
Several polynomials associated with the harmonic numbers
Discrete Applied Mathematics
A generalization of Fibonacci and Lucas matrices
Discrete Applied Mathematics
Note: Identities via Bell matrix and Fibonacci matrix
Discrete Applied Mathematics
Several identities for the generalized Apostol-Bernoulli polynomials
Computers & Mathematics with Applications
Bernoulli polynomials and Pascal matrices in the context of Clifford analysis
Discrete Applied Mathematics
Note: Ballot matrix as Catalan matrix power and related identities
Discrete Applied Mathematics
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In this paper, we define the generalized Bernoulli polynomial matrix B(x)(x) and the Bernoulli matrix B. Using some properties of Bernoulli polynomials and numbers, a product formula of B(x)(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) = F M/(x) = N(x)J, 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) Bi-j(x) - (i-1 j) Bi-j-1(x) - (i-2 j) Bi-j-2(x)) and ((i j) Bi-j(x) - (i j+1) Bi-j-1(x) - (i j+2)Bi-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.