Related papers: Integer powers of certain complex tridiagonal matr…

It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…

We study complex interpolation of variable Triebel-Lizorkin spaces, especially we present the complex interpolation of $F_{p(\cdot),q}^{\alpha }$ and $F_{p(\cdot ),p(\cdot )}^{\alpha (\cdot )}$ spaces. Also, some limiting cases are given.

In this paper, we give quadratic approximation of generalized Tribonacci sequence $\{V_{n}\}_{n\geq0}$ defined by Eq. (\ref{eq:7}) and use this result to give the matrix form of the $n$-th power of a companion matrix of…

We introduce three simple polynomial maps with integer coefficients that have interesting dynamical properties modulo primes.

In this paper, as an analogue of the integer case, we study detailedly the period and the rank of the generalized Fibonacci sequence of polynomials over a finite field modulo an arbitrary polynomial. We establish some formulas to compute…

Factorization of an $n\times n$ unitary matrix as a product of $n$ diagonal matrices containing only phases interlaced with $n-1$ orthogonal matrices each one generated by a real vector as well as an explicit form for the Weyl factorization…

In this paper, we present several explicit formulas of the sums and hyper-sums of the powers of the first (n+1)-terms of a general arithmetic sequence in terms of Stirling numbers and generalized Bernoulli polynomials.

For an arbitrary finite set S of natural numbers greater 1, we construct an integer-valued polynomial f, whose set of lengths in Int(Z) is S. The set of lengths of f is the set of all natural numbers n, such that f has a factorization as a…

A QR factorization of a tall and skinny matrix with n columns can be represented as a reduction. The operation used along the reduction tree has in input two n-by-n upper triangular matrices and in output an n-by-n upper triangular matrix…

Given an odd prime $p$, we provide formulas for the Hensel lifts of polynomial roots modulo $p$, and give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose…

We present a lovely connection between the Fibonacci numbers and the sums of inverses of $(0,1)-$ triangular matrices, namely, a number $S$ is the sum of the entries of the inverse of an $n \times n$ $(n \geq 3)$ $(0,1)-$ triangular matrix…

The general formulas for finding the quantity of all primitive and nonprimitive triples generated by the given number x have been proposed. Also the formulas for finding the complete quantity of the representations of the integers as a…

Using modular forms we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.

Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.

For an integer $ k\geq 2 $, let $ \{F^{(k)}_{n} \}_{n\geq 0}$ be the $ k$--generalized Fibonacci sequence which starts with $ 0, \ldots, 0, 1 $ ($ k $ terms) and each term afterwards is the sum of the $ k $ preceding terms. In this paper,…

We give some new canonical representations for forms over $\cc$. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can…

For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n}$ be the $k-$Fibonacci sequence which starts with $0,\ldots,0,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we search for powers of 2 which are…

For an integer $k\geq 2$, let $\{F^{(k)}_{n}\}_{n\geqslant 2-k}$ be the $ k$--generalized Fibonacci sequence which starts with $0, \ldots, 0,1$ (a total of $k$ terms) and for which each term afterwards is the sum of the $k$ preceding terms.…

In this paper, we find all integers $c$ having at least two representations as a difference between a Fibonacci number and a Tribonacci number.

We give the explicit analytic development of Macdonald polynomials in terms of "modified complete" and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments…