Related papers: Linear divisibility sequences and Salem numbers

Let $\theta $ be a Salem number and $P(x)$ a polynomial with integer coefficients. It is well-known that the sequence $(\theta^n)$ modulo 1 is dense but not uniformly distributed. In this article we discuss the sequence $(P(\theta^n))$…

Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials…

Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations and third order quintically semi-linear ordinary differential equations, we extend to the fourth order by differentiating…

We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider…

We study the class of those linear relations that can be factorized as products of idempotent relations. We provide several characterizations of this class, extending known factorization results for operators to the more general setting of…

We study the Coxeter polynomials associated with certain star-like trees. In particular, we exhibit large Salem factors of these polynomials and give convergence properties of their dominant roots.

In this paper we consider divisibility sequences obtained from square matrices. We work with of matrix divisibility sequences associated to a semigroup and arising from endomorphisms of an affine space. We prove that determinant…

We study sets of integers that can be defined by the vanishing of a generalised polynomial expression. We show that this includes sets of values of linear recurrent sequences of Salem type and some linear recurrent sequences of Pisot type.…

Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and successive derivatives of binomial type sequences. We give some relations between Bell polynomials and…

We show that many existing divisibility sequences can be seen as sequences of determinants of matrix divisibility sequences, which arise naturally as Jacobian matrices associated to groups of maps on affine spaces.

We classify all linear division sequences in the integers, a problem going back to at least the 1930s. As a corollary we also classify those linear recurrence sequences in the integers for which $(x_m,x_n)=\pm x_{(m,n)}$. We also show that…

In this paper, we consider several special polynomials related to associated sequences of polynomials. Finally, we give some new and interesting identities of those polynomials arising from transfer formula for the associated sequences.

In this note, we study the divisibility relation $U_m\mid U_{n+k}^s-U_n^s$, where ${\bf U}:=\{U_n\}_{n\ge 0}$ is the Lucas sequence of characteristic polynomial $x^2-ax\pm 1$ and $k,m,n,s$ are positive integers.

For a sequence of polynomials $\{p_k(t)\}$ in one real or complex variable, where $p_k$ has degree $k$, for $k\ge 0$, we find explicit expressions and recurrence relations for infinite matrices whose entries are the coefficients $d(n,m,k)$,…

We investigate strong divisibility sequences and produce lower and upper bounds for the density of integers in the sequence which only have (somewhat) large prime factors. We focus on the special cases of Fibonacci numbers and elliptic…

Consider a sequence of real-valued functions of a real variable given by a homogeneous linear recursion with differentiable coefficients. We show that if the functions in the sequence are differentiable, then the sequence of derivatives…

We define the $\textit{Divisor Divisibility Sequence}$ associated to a Laurent polynomial $f\in\mathbb{Z}[X_1^{\pm1},\ldots,X_N^{\pm1}]$ to be the sequence $W_n(f)=\prod f(\zeta_1,\ldots,\zeta_N)$, where $\zeta_1,\ldots,\zeta_N$ range over…

We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable…

Generalized cyclotomic sequences of period pq have several desirable randomness properties if the two primes p and q are chosen properly. In particular,Ding deduced the exact formulas for the autocorrelation and the linear complexity of…

In this article, a new approach based on linear algebra is adopted to study a hybrid Sheffer polynomial sequences. The recurrence relations and differential equation for these polynomials are derived by using the properties and…