Netted Binomial Matrices

The duality triads were defined in the preceding paper.(ArXiv: math.GM/0402260 v 1 Feb. 2004). Notation, enumeration of formulas and references is therefore to be continued hereby. In this paper Fibonomial triangle and further Pascal-like triangles including q-Gaussian one are given explicit interpretation as discrete time dynamical systems as it is the case with all duality triads.

In this paper we study the action of the Binomial and Invert (interpolated) operators on the set of linear recurrent sequences. We prove that these operators preserve this set, and we determine how they change the characteristic polynomials. We show that these operators, with the aid of two other elementary operators (essentially the left and right shifts), can transform any impulse sequence (a linear recurrent sequence starting from $(0,...,0,1)$) into any other impulse sequ...

The research aims to construct a new type of matrix called the Fibonacci-Hessenberg-Lorentz matrix by multiplying Fibonacci-Hessenberg matrices with Lorentz matrix multiplication. The study will start by examining the properties of Hessenberg and tridiagonal matrices and then focus on developing the Fibonacci-Hessenberg matrix using Fibonacci sequences. By multiplication it with a Lorentz matrix multiplication, the resulting matrix, the Fibonacci-Hessenberg-Lorentz matrix, wi...

We examine relationships between two minors of order n of some matrices of n rows and n+r columns. This is done through a class of determinants, here called $n$-determinants, the investigation of which is our objective. We prove that 1-determinants are the upper Hessenberg determinants. In particular, we state several 1-determinants each of which equals a Fibonacci number. We also derive relationships among terms of sequences defined by the same recurrence equation independen...

A technique to compute arbitrary products of a class of Fibonacci $2\times2$ square matrices is proved in this work. General explicit solutions for non autonomous Fibonacci difference equations are obtained from these products. In the periodic non autonomous Fibonacci difference equations the monodromy matrix, the Floquet multipliers and the Binet's formulas are obtained. In the periodic case explicit solutions are obtained and the solutions are analyzed.

In this paper, by some arithmetic properties of the Pell sequence and some $p$-adic tools, we study certain cyclotomic matrices involving squares over finite fields. For example, let $1=s_1,s_2,\cdots,s_{(q-1)/2}$ be all the nonzero squares over $\mathbb{F}_{q}$, where $q=p^f$ is an odd prime power with $q\ge7$. We prove that the matrix $$B_q((q-3)/2)=\left[\left(s_i+s_j\right)^{(q-3)/2}\right]_{2\le i,j\le (q-1)/2}$$ is a singular matrix whenever $f\ge2$. Also, for the c...

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 produ...

Combinatorial interpretation of the fibonomial coefficients recently proposed by the present author results here in combinatorial interpretation of the recurrence relation for fibonomial coefficients . The presentation is provided with quite an exhaustive context where apart from plane grid coordinate system used several figures illustrate the exposition of statements and the derivation of the recurrence itself.

One of the most interesting results of the last century was the proof completed by Matijasevich that computably enumerable sets are precisely the diophantine sets [MRDP Theorem, 9], thus settling, based on previously developed machinery, Hilbert's question whether there exists a general algorithm for checking the solvability in integers of any diophantine equation. In this paper we describe techniques to prove the nonexistence of polynomials in two variables for some simple g...

In this paper, we derive the general expression of the r-th power for some n-square complex tridiagonal matrices. Additionally, we obtain the complex factorizations of Fibonacci polynomials.