Determinant Formulas of Some Hessenberg Matrices with Jacobsthal Entries (original) (raw)

Determinant Identities for Toeplitz-Hessenberg Matrices with Tribonacci Number Entries

arXiv: Combinatorics, 2020

In this paper, we evaluate determinants of some families of Toeplitz-Hessenberg matrices having tribonacci number entries. These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers. In particular, we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz-Hessenberg determinants. We then obtain, by combinatorial arguments, extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying an r-th order recurrence of a more general form with the appropriate initial conditions, where r>2 is arbitrary.

A unified approach for the Hankel determinants of classical combinatorial numbers

Journal of Mathematical Analysis and Applications, 2015

We give a general formula for the determinants of a class of Hankel matrices which arise in combinatorics theory. We revisit and extend existant results on Hankel determinants involving the sum of consecutive Catalan, Motzkin and Schroder numbers and we prove a conjecture in [20] about the recurrence relations satisfied by the Hankel transform of linear combinations of Catalans numbers.

Determinantal Identities for K Lucas Sequence

2016

In this paper, we de¯ned new relationship between k Lucas sequences and determinants of their associated matrices, this approach is di®erent and never tried in k Fibonacci sequence literature.

Determinants of Toeplitz–Hessenberg matrices with generalized Fibonacci entries

Notes on Number Theory and Discrete Mathematics

In this paper, we evaluate several families of Toeplitz-Hessenberg matrices whose entries are generalized Fibonacci numbers. In particular, we find simple formulas for several determinants whose entries are translates of the Chebyshev polynomials of the second kind. Equivalently, these determinant formulas may also be rewritten as identities involving sums of products of generalized Fibonacci numbers and multinomial coefficients. Combinatorial proofs which make use of sign-reversing involutions and the definition of a determinant as a signed sum over the symmetric group S n are given for our formulas in several particular cases, including those involving the Chebyshev polynomials.

Evaluation of Hessenberg determinants via generating function approach

Filomat, 2017

In this paper, we will present various results on computing of wide classes of Hessenberg matrices whose entries are the terms of any sequence. We present many new results on the subject as well as our results will cover and generalize earlier many results by using generating function method. Moreover, we will present a new approach on computing Hessenberg determinants, whose entries are general higher order linear recursions with arbitrary constant coefficients, based on finding an adjacency-factor matrix. We will give some interesting showcases to show how to use our new method.

On the recurrences of the Jacobsthal sequence

Mathematica, 2023

In the present work, two new recurrences of the Jacobsthal sequence are defined. Some identities of these sequences which we call the Jacobsthal array is examined. Also, the generating and series functions of the Jacobsthal array are obtained. MSC 2020. 11B39, 05A15, 11B83.

Jacobsthal Numbers and Associated Hessenberg Matrices

J. Integer Seq., 2018

In this paper, we define two n × n Hessenberg matrices, one of which corresponds to the adjacency matrix of a bipartite graph. We then investigate the relationships between the Hessenberg matrices and the Jacobsthal numbers. Moreover, we give Maple algorithms to verify our results.

Determinant formulas of some Toeplitz–Hessenberg matrices with Catalan entries

Proceedings - Mathematical Sciences, 2019

In this paper, we consider determinants of some families of Toeplitz-Hessenberg matrices having various translates of the Catalan numbers for the nonzero entries. These determinant formulas may also be rewritten as identities involving sums of products of Catalan numbers and multinomial coefficients. Combinatorial proofs may be given for several of the identities that are obtained.