Some Infinite Matrices Whose Leading Principal Minors Are Well-known Sequences

There are scattered results in the literature showing that the leading principal minors of certain infinite integer matrices form the Fibonacci and Lucas sequences. In this article, among other results, we have obtained new families of infinite matrices such that the leading principal minors of them form a famous integer (sub)sequence, such as Fibonacci, Lucas, Pell and Jacobsthal (sub)sequences.Comment: 20 pages, Utilitas Mathematica, 201

Matrices in the Hosoya triangle

In this paper we use well-known results from linear algebra as tools to explore some properties of products of Fibonacci numbers. Specifically, we explore the behavior of the eigenvalues, eigenvectors, characteristic polynomials, determinants, and the norm of non-symmetric matrices embedded in the Hosoya triangle. We discovered that most of these objects either embed again in the Hosoya triangle or they give rise to Fibonacci identities. We also study the nature of these matrices when their entries are taken $\bmod$ $2$. As a result, we found an infinite family of non-connected graphs. Each graph in this family has a complete graph with loops attached to each of its vertices as a component and the other components are isolated vertices. The Hosoya triangle allowed us to show the beauty of both, the algebra and geometry.Comment: Six figure

Determinants containing powers of polynomial sequences

We derive identities for the determinants of matrices whose entries are (rising) powers of (products of) polynomials that satisfy a recurrence relation. In particular, these results cover the cases for Fibonacci polynomials, Lucas polynomials and certain orthogonal polynomials. These identities naturally generalize the determinant identities obtained by Alfred, Carlitz, Prodinger, Tangboonduangjit and Thanatipanonda.Comment: 12 page

Determinants Containing Powers of Generalized Fibonacci Numbers

We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These studies have led us to discover a fundamental identity of determinant involving powers of linear polynomials. Finally, we discuss the determinants of matrices whose entries are products of the generalized Fibonacci numbers

On the Determinants and Inverses of Circulant Matrices with a General Number Sequence

The generalized sequence of numbers is defined by W_{n}=pW_{n-1}+qW_{n-2} with initial conditions W_{0}=a and W_{1}=b for a,b,p,q\inZ and n\geq2, respectively. Let W_{n}=circ(W_{1},W_{2},...,W_{n}). The aim of this paper is to establish some useful formulas for the determinants and inverses of W_{n} using the nice properties of the number sequences. Matrix decompositions are derived for W_{n} in order to obtain the results

Determinants and Inverses of Circulant Matrices with Jacobsthal and Jacobsthal-Lucas Numbers

Let n\geq3 and J_{n}:=circ(J_{1},J_{2},...,J_{n}) and j_{n}:=\circ(j_{0},j_{1},...,j_{n-1}) be the n\timesn circulant matrices, associated with the nth Jacobsthal number J_{n} and the nth Jacobsthal-Lucas number j_{n}, respectively. The determinants of J_{n} and j_{n} are obtained in terms of the Jacobsthal and Jacobsthal-Lucas numbers. These imply that J_{n} and j_{n} are invertible. We also derive the inverses of J_{n} and j_{n}

Explicit formulae for spectral norms of circulant-type matrices with some given entries

In this paper we investigate the spectral norm for circulant matrices, whose entries are modified Fibonacci numbers and Lucas numbers. We obtain the identity estimations for the spectral norms. Some numerical test results are listed to verify the results using those approaches.Comment: 7 page

Permanents, Determinants, Weighted Isobaric Polynomials and Integer Sequences

In this paper we construct two types of Hessenberg matrices with the properties that every weighted isobaric polynomial (WIP) appears as a determinant of one of them, and as the permanent of the other. Every integer sequence which is linearly recurrent is representable by (an evaluation of) some linearly recurrent sequence of WIPs. WIPs are symmetric polynomials written on the elementary symmetric polynomial basis. Among them are the generalized Fibonacci polynomials and the generalized Lucas polynomials, which already have these sweeping representing properties. Among the integer sequences discussed are the Chebychev polynomials of the 2nd kind, the Stirling numbers of the 1st and 2nd kind, the Catalan numbers, and the triangular numbers, as well as all sequences which are either multiplicative arithmetic functions or additive arithmetic functions

The Hadamard Products for bi-periodic Fibonacci and bi-periodic Lucas Generating matrices

In this paper, firstly, we define the Qq-generating matrix for bi-periodic Fibonacci polynomial. And we give nth power, determinant and some properties of the bi-periodic Fibonacci polynomial by considering this matrix representation. Also, we introduce the Hadamard products for bi-periodic Fibonacci Qnq generating matrix and bi-periodic Lucas Qnl generating matrix of which entries is bi-periodic Fibonacci and Lucas numbers. Then, we investigate some properties of these products

Star of David and other patterns in the Hosoya-like polynomials triangles

In this paper we first generalize the numerical recurrence relation given by Hosoya to polynomials. Using this generalization we construct a Hosoya-like triangle for polynomials, where its entries are products of generalized Fibonacci polynomials (GFP). Examples of GFP are: Fibonacci polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell polynomials, Fermat polynomials, Jacobsthal polynomials, Vieta polynomials and other familiar sequences of polynomials. For every choice of a GFP we obtain a triangular array of polynomials. In this paper we extend the star of David property, also called the Hoggatt-Hansell identity, to this type of triangles. We also establish the star of David property in the gibonomial triangle. In addition, we study other geometric patterns in these triangles and as a consequence we give geometric interpretations for the Cassini's identity, Catalan's identity, and other identities for Fibonacci polynomials.Comment: Eight Figures and 24 page