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Mohammed Mouçouf

Publications and source records attributed to Mohammed Mouçouf.

9 recordsLinked to original sources
arXiv2024

Some algebraic results concerning linear recurrence sequences

We study the set $\mathcal{L}_{F}$ of all $F$-vector spaces $L(P)$ where $P$ is monic and splits over $F$ and $L(Q)$ denotes the set of linear recurrence sequences over $F$ with characteristic polynomial $Q$. We show that $\mathcal{L}_{F}$ can be endowed with two structures of graded commutative semiring. This study allows us to obtain, in compact forms, the polynomial $P,Q\in F[X]$ such that $L(P)=\prod_{i=1}^{m}L(P_{i})$ and $L(Q)=L(P_{1})\ast\cdots\ast L(P_{m})$, where $P_{1}, \ldots, P_{m}$ are any monic polynomials over $F$.

math.RA↗
arXiv2023

A common generalization of Dickson polynomials, Fibonacci polynomials, and Lucas polynomials and applications

In this work, we define a more general family of polynomials in several variables satisfying a linear recurrence relation. Then we provide explicit formulas and determinantal expressions. Finally, we apply these results to recurrent polynomials of order $2$, we present several relations and interesting identities involving the Fibonacci polynomials of order $2$, the Lucas polynomials of order $2$, the classical Fibonacci polynomials, the classical Lucas polynomials, the Fibonacci numbers, the Lucas numbers, the Dickson polynomials of the first kind, and the Dickson polynomials of the second kind. Our results are a unified generalization of several works. Some well known results are special cases of ours.

math.NT↗
arXiv2023

Explicit, recurrent, determinantal expressions of the $k$th power of formal power series and applications to the generalized Bernoulli numbers

In this work, the authors provide closed forms and recurrence expressions for computing the $k$th power of the formal power series, some of them in terms of a determinant of some matrices. As a consequence, we obtain the reciprocal of the unit of any formal power series. We apply these results to the generalized Bernoulli numbers and Bernoulli numbers, we derive new closed-form expressions and some recursive relations of these famous numbers. In addition, we present several identities in determinant form. Using these results, an elegant generalization of a well known identity of Euler is presented. We also note some connections between the Stirling numbers of the second kind and the generalized Bernoulli numbers.

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arXiv2023

Algebraic identities on q-harmonic numbers and q-binomial coefficients

The aim of this paper is to present a general algebraic identity. Applying this identity, we provide several formulas involving the q-binomial coefficients and the q-harmonic numbers. We also recover some known identities including an algebraic identity of D. Y. Zheng on q-Apéry numbers and we establish the q-analog of Euler's formula. The proposed results may have important applications in the theory of q-supercongruences.

math.CO↗
arXiv2021

An explicit expression for the minimal polynomial of the Kronecker product of matrices. Explicit formulas for matrix logarithm and matrix exponential

Using $\mathcal{P}$-canonical forms of matrices, we derive the minimal polynomial of the Kronecker product of a given family of matrices in terms of the minimal polynomials of these matrices. This, allows us to prove that the product $\prod\limits_{i=1}^{m}L(P_{i})$, $L(P_{i})$ is the set of linear recurrence sequences over a field $F$ with characteristic polynomial $P_{i}$, is equal to $L(P)$ where $P$ is the minimal polynomial of the Kronecker product of the companion matrices of $P_{i}$, $1\leq i\leq m$. Also, we show how we deduce from the $\mathcal{P}$-canonical form of an arbitrary complex matrix $A$, the $\mathcal{P}$-canonical form of the matrix function $e^{tA}$ and a logarithm of $A$.

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arXiv2021

Some Explicit Formulas for Matrix Exponential, Matrix Logarithm, the $n$th Power of Matrices and their Drazin Inverses

In this work, new closed-form formulas for the matrix exponential are provided. Our method is direct and elementary, it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Moreover, others are recuperated and generalized. As a consequence, we easily obtain the Chevalley{Jordan decomposition and the spectral projections of any matrix. In addition, closed-form expressions for the arbitrary positive powers of matrices and their Drazin inverses are presented. Using these results, an elegant explicit formula for logarithm of matrices is obtained. Several particular cases and examples are formulated to illustrate the methods presented in this paper.

math.RA↗
arXiv2020

P-canonical forms and complete inverses

This paper describes a new kind of inverse for elements in associative ring, that is the complete inverse, as the unique solution of a certain set of equations. This inverse exists for an element $a$ if and only if the Drazin inverse of $a$ exists. We also show that by plugging in $-k$ for $k$ in the $\mathcal{P}$-canonical form of a square matrix $A$, we get the $\mathcal{P}$-canonical form of the complete inverse of $A$.

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arXiv2020

S-rings of Krull type

We define and give some properties and characterizations of S-rings of Krull type. We also determine, in the case of an independent S-ring of Krull type A, the injective dimension of the quotient category Mod(A)/\mathcal{M}_{0}, where \mathcal{M}_{0} is the thick subcategory of the modules with trivial maps into the codivisorial modules.

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