Pascal Matrix Research Articles

The linear algebra of the generalized Pascal matrix with four variables

The linear algebra of the generalized Pascal matrix with four variables

A low-time-consumption image encryption combining 2D parametric Pascal matrix chaotic system and elementary operation

The rapid development of the big data era has resulted in traditional image encryption algorithms consuming more time in handling the huge amount of data. The consumption of time cost needs to be reduced while ensuring the security of encryption algorithms. With this in mind, the paper proposes a low-time-consumption image encryption (LTC-IE) combining 2D parametric Pascal matrix chaotic system (2D-PPMCS) and elementary operation. First, the 2D-PPMCS with robustness and complex chaotic behavior is adopted. Second, the SHA-256 hash values are applied to the chaotic sequences generated by 2D-PPMCS, which are processed and applied to image permutation and diffusion encryption. In the permutation stage, the pixel matrix is permutation encrypted based on the permutation matrix generated from the chaotic sequences. For diffusion encryption, elementary operations are utilized to construct the model, such as exclusive or, modulo, and arithmetic operations (addition, subtraction, multiplication, and division). After analyzing the security experiments, the LTC-IE algorithm ensures security and robustness while reducing the time cost consumption.

A study on some paranormed sequence spaces due to Lambda–Pascal matrix

A study on some paranormed sequence spaces due to Lambda–Pascal matrix

The k-Tribonacci Matrix and the Pascal Matrix

This article discusses the relationship between the k-Tribonacci matrix Tn(k) and the Pascal matrix Pn, by first constructing the k-Tribonacci matrix and then looking for its inverse. From the inverse k-Tribonacci matrix, unique characteristics can be constructed so that general shapes can be constructed, and then from the relationship of the k-Tribonacci matrix Tn(k) and the Pascal matrix Pn obtain a new matrix, i.e. Un(k). Furthermore, a factor is derived from the relationship of the k-Tribonacci matrix Tn(k) and the Pascal matrix Pn i.e. Pn = Tn(k)Un(k).

New Results Parametric Apostol-Type Frobenius-Euler Polynomials and their Matrix Approach.

The new algebraic properties of the parametric Apostol-type Frobenius-Euler polynomials and parametric type Frobenius-Euler polynomial have been explained in this research. The researchers have studied the series of the Taylor type and established the relation between the classic Apostol Frobenius-Euler and Frobenius-Euler polynomials. This work has also addressed the generalized parametric Apostol-type Frobenius-Euler polynomials matrices and has shown some of their properties. Finally, this research provided some factorizations of Apostol-type Frobenius-Euler matrix that involves the generalized Pascal matrix, Fibonacci and Lucas matrices, respectively.

ON GENERALIZATIONS OF STIRLING NUMBERS AND SOME WELL-KNOWN MATRICES

We introduce a generalization of the Stirling numbers of the first kind and the second kind. By arranging these numbers into matrices, we generalize the Stirling matrices of the first kind and the second kind investigated by Cheon and Kim [Stirling matrix via Pascal matrix, Linear Algebra Appl. 329 (2001) 49–59]. Furthermore, we introduce generalizations of the Pascal matrix and the symmetric Pascal matrix with two real arguments, and generalize earlier results related to the Pascal matrices, Stirling matrices and matrices involving Bell numbers.

The Linear Algebra of (r, β)-Stirling Matrices

This paper establishes the linear algebra of the $(r, \beta)$-Stirling matrix. Along the way, this paper derives various identities, such as its factorization and relationship to the Pascal matrix and the Stirling matrix of the second kind. Additionally, this paper develops a natural extension of the Vandermonde matrix, which can be used to study and evaluate successive power sums of arithmetic progressions.

A Look at Generalized Degenerate Bernoulli and Euler Matrices

In this paper, we consider the generalized degenerate Bernoulli/Euler polynomial matrices and study some algebraic properties for them. In particular, we focus our attention on some matrix-inversion formulae involving these matrices. Furthermore, we provide analytic properties for the so-called generalized degenerate Pascal matrix of the first kind, and some factorizations for the generalized degenerate Euler polynomial matrix.

D-Gaussian Fibonacci,d-Gaussian Lucas polynomials and their matrix representations

UDC 517.5 We defined-Gaussian Fibonacci polynomials andd-Gaussian Lucas polynomials. We present the matrix representations of these polynomials. By using the Riordan method, we obtain the factorizations of the Pascal matrix including the polynomials. In addition, we define the infinited-Gaussian Fibonacci polynomial matrix and thed-Gaussian Lucas polynomial matrix and give their inverses.

D-Tribonacci Polynomials and Their Matrix Representations

In this study, we define d-Tribonacci polynomials. Some combinatorial properties of the d- Tribonacci polynomials with matrix representations are obtained with the help of Riordan arrays. In addition, d- Tribonacci number sequence, which is a new generalization of this number sequence, has been obtained by considering Pascal matrix. With the help of Pascal matrix, two kinds of factors of d-Tribonacci polynomials were found. Also, infinite d-Tribonacci polynomials matrix and the inverses of these polynomials were found.

A generalization of the pascal matrix and an application to coding theory

Suppose that m , k , t are integers with m , k ≥ 1 and 0 ≤ t and A is the k × k matrix with the ( i , j ) -entry ( t + ( j − 1 ) m i − 1 ) . When t = 0 and m = 1, this is the upper triangular Pascal matrix. Here, first we study the properties of this matrix, in particular, we find its determinant and its LDU decomposition and also study its inverse. Then by using this matrix we present a generalization of the Mattson-Solomon transform and a polynomial formulation for its inverse to the case that the sequence length and the characteristic of the base field are not coprime. At the end, we use this generalized Mattson-Solomon transform to present a lower bound on the length of repeated root cyclic codes, which can be seen as a generalization of the BCH bound.

An image encryption scheme based on a seven-dimensional hyperchaotic system and Pascal's matrix

An image encryption scheme based on a seven-dimensional hyperchaotic system and Pascal's matrix

The skew halves of a Riordan array

Given a Riordan array (gn,k)n,k∈N, its vertical half (g2n−k,n)n,k∈N and horizontal half (g2n,n+k)n,k∈N are studied separately before. In the present paper, we introduce the skew (r,s)-halves of a Riordan array which are infinite lower triangular matrices with generic (n,k)-th entries g2n+(s−2)k+r,n+(s−1)k+r for n≥k≥0. This allows us to discuss the vertical half and horizontal half in a uniform context. We show that the skew halves of a Riordan array are all Riordan arrays. As applications, we find several new identities involving the Pascal matrix, and Catalan triangles by applying the skew halves. We also consider the inversion problems: given a Riordan array G, we can construct its Riordan antecedent H such that the (r,s)-half of H is equal to G.

Some binomial identities related to the Catalan triangles and the halves of the Pascal matrix

In this paper, five binomial sums with two additional parameters are shown to be equipollent by using the Riordan array method and the generalized Catalan matrices, as well as the halves of the Pascal matrix.

D-Gaussian Pell-Lucas Polynomials and Their Matrix Representations

We define a new generalization of Gaussian Pell-Lucas polynomials. We call it $d-$Gaussian Pell-Lucas polynomials. Then we present the generating function and Binet formula for the polynomials. We give a matrix representation of $d-$Gaussian Pell-Lucas polynomials. Using the Riordan method, we obtain the factorizations of Pascal matrix involving the polynomials.

An Application of the t -Extension of the p -Fibonacci Pascal Matrix in Coding Theory

We consider the t -extension of the p -Fibonacci Pascal matrix. First, we study the k -th power of the t -extension of the p -Fibonacci lower and upper triangular Pascal matrix. Then, we obtain a new code which is named the t -extension of the p -Fibonacci Pascal matrix coding/decoding by using them.

D-Gaussian Pell polynomials and their matrix representation

We introduce a new generalization of Gaussian Pell polynomials. We call it [Formula: see text]-Gaussian Pell polynomials. Then we present the sum, generating functions and Binet formulas of these polynomials. We give the matrix representation of [Formula: see text]-Gaussian Pell polynomials. We introduce the matrix as binary representations according to the Riordan matrix representation. Using the Riordan method, we obtain the factorizations of the Pascal matrix involving the polynomial. In addition, we present the inverse of matrices of the polynomial.

On some Lambda–Pascal sequence spaces and compact operators

We introduce Lambda–Pascal sequence spaces ℓq(G), c0(G), c(G) and ℓ∞(G) generated by the matrix G which is obtained by the product of Pascal matrix and Λ-matrix. It is proved that the Lambda–Pascal sequence spaces ℓq(G), c0(G), c(G) and ℓ∞(G) are BK-spaces and linearly isomorphic to ℓq, c0, c and ℓ∞, respectively. We construct Schauder bases and obtain α-, β- and γ-duals of the new spaces. We state and prove characterization theorems related to matrix transformation from the space ℓq(G) to the spaces ℓ∞, c and c0. Finally, we determine necessary and sufficient conditions for a matrix operator to be compact from the space c0(G) to any one of the spaces ℓ∞, c, c0 or ℓ1.

The Jordan and Frobenius pairs of the inverse

Given a matrix there exists a nonsingular matrix such that , where J is a very sparse matrix with a diagonal block structure, known as the Jordan canonical form (JCF) of . Assume that is nonsingular and that and are given. How to obtain and such that and is the JCF of ? Curiously, the answer involves the Pascal matrix. For the Frobenius canonical form (FCF), where blocks are companion matrices, the analogous question has a very simple answer. Jordan blocks and companions are non-derogatory lower Hessenberg matrices. The answers to the two questions will be obtained by solving two linear matrix equations involving these matrices.

Factorizations of some lower triangular matrices and related combinatorial identities

In this study, we investigate the connection between second order recurrence matrix and several combinatorial matrices such as generalized r-eliminated Pascal matrix, Stirling matrix of the first and of the second kind matrices. We give factorizations and inverse factorizations of these matrices by virtue of the second order recurrence matrix. Moreover, we derive several combinatorial identities which are more general results of some earlier works.