Related papers: New examples of determinant divisibility sequences
A novel kind of self-referential square matrix is introduced. A certain subset of the matrix entries record the frequencies of occurrence of each distinct number appearing within the entire matrix. Such squares are necessarily elusive. Our…
In this paper, we find the sums in closed form of certain type of Lucas-related convergent series. More precisely, we generalize the results already obtained by the author in his arXiv paper entitled: "Summation of certain infinite…
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
It is known that every complex square matrix with nonnegative determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product.…
A lucasene is a hexagon chain that is similar to a fibonaccene, an $L$-fence is a poset the Hasse diagram of which is isomorphic to the directed inner dual graph of the corresponding lucasene. A new class of cubes, which named after…
A second order polynomial sequence is of \emph{Fibonacci-type} (\emph{Lucas-type}) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Known examples of these type of sequences are: Fibonacci polynomials,…
We investigate the generalized moment membership problem for matrices, a formulation equivalent to Skolem's problem for linear recurrence sequences. We show decidability for orthogonal, unitary, and real eigenvalue matrices, and…
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 $\mathcal{L}_{F}$…
In this paper, in the first we give definitions of some classes of division rings which strictly contain the class of centrally finite division rings. One of our main purpose is to construct non-trivial examples of rings of new defined…
We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the…
This paper deals with new sequence spaces $X(r, s, t ;\Delta) $ for $X\in \{l_\infty, c, c_0\}$ defined by using generalized means and difference operator. It is shown that these spaces are complete normed linear spaces and the spaces $X(r,…
We present a different proof of the characterization of non--degenerate recurrence sequences, which are also divisibility sequences, given by Van der Poorten, Bezevin, and Petho in their paper "A Full Characterisation of Divisibility…
In this paper, we study a structured family of matrices whose entries are given by products of $k$-Fibonacci and $k$-Lucas numbers. For this family, we obtain explicit and unified formulas for several classical matrix invariants, including…
The goal of this paper is to present an algebraic approach to the basic results of the theory of linear recurrence relations. This approach is based on the ideas from the theory of representations of one endomorphisms (a special case of…
One of the basic problems in studying topological structures of deformation spaces for Kleinian groups is to find a criterion to distinguish convergent sequences from divergent sequences. In this paper, we shall give a sufficient condition…
Using generalized binomial coefficients with respect to fundamental Lucas sequences we establish congruences that generalize the classical congruence of Wolstenholme and other related stronger congruences.
A divisibility sequence is a sequence of integers $\{d_n\}$ such that $d_m$ divides $d_n$ if $m$ divides $n$. Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to…
In this paper a general theory of semi-classical matrix orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distributional equation $D(u A) = u B,$ where $A$ and $B$ are matrix polynomials.…
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 derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.