Related papers: A New Generalized Cassini Determinant
A new matrix operation based on inserting columns and rows, similarly to the mediant operation between fractions, gives rise to the Farey determinants matrix or, equivalently, the matrix of the numerators of the differences of Farey…
A general discussion of the Cabibbo-Kobayashi-Maskawa (CKM) matrix is given and the importance stressed of determining the matrix elements as an essential part of understanding CP violation in and beyond the Standard Model. The status of…
In this article, we will discover some new generalized identity regarding continued fractions. We will connect the results to Fibonacci numbers and Lucas numbers. For all the proof, we will use induction.
This paper presents new identities expressing the terms of Fibonacci, Lucas, and generalized Fibonacci sequences with multiple indices through powers of Lucas numbers and binomial coefficients. The obtained formulas rely on the application…
We establish new connection formulae between Fibonacci polynomials and Chebyshev polynomials of the first and second kinds. These formulae are expressed in terms of certain values of hypergeometric functions of the type 2F1. Consequently,…
Quasi-cyclic (QC) codes form an important generalization of cyclic codes. It is well know that QC codes of length $s\ell$ with index $s$ over the finite field $\mathbb{F}$ are $\mathbb{F}[y]$-submodules of the ring $\frac{\mathbb{F}[x,y]}{<…
We give an elementary proof of a Caratheodory-type result on the invertibility of a sum of matrices, due first to Facchini and Barioli. The proof yields a polynomial identity, expressing the determinant of a large sum of matrices in terms…
Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, we define a new polynomial related to the higher-order generalized…
Up-down permutations are counted by tangent resp. secant numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
The Riccati equation method is used to establish new oscillation criteria for extended linear matrix Hamiltonian systems. This method allows to obtain results in in a new direction, which is to break the positive definiteness condition,…
We show how to construct central and grouplike quantum determinants for FRT algebras A(R). As an application of the general construction we give a quantum determinant for the q-Lorentz group.
In this paper, we define new generalized k-Mersenne numbers and give a formula of generalized Mersenne polynomials and further we study their properties. Moreover, we define Gaussian Mersenne numbers and obtain some identities like Binet…
This paper generalizes a graph theoretic proof technique for a Fibonacci identity proposed by Lee Knisley Sanders, and explores characteristics of these generalized theorems ad infinitum.
In this article, we introduce and study a new integer sequence referred to as the higher order Mersenne sequence. The proposed sequence is analogous to the higher order Fibonacci numbers and closely associated with the Mersenne numbers.…
We give new definitions for the determinant over commutative ring $K$, noncommutative ring $\mathbf{K}$, noncommutative ring $\mathcal{K}$ with associative powers, over noncommutative nonassociative ring $\mathfrak{K}$, and study their…
A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.
We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…
After analyzing the 4x4 determinant of a matrix, a shortcut was obtained to find such a determinant. Similarly to the Sarrus method for 2x2 or 3x3 determinants, the method consists of laying 19 columns of size 4 each and adding and…
Two well-known $q$-Hermite polynomials are the continuous and discrete $q$-Hermite polynomials. In this paper we consider a new family of $q$-Hermite polynomials and prove several curious properties about these polynomials. One striking…