Related papers: Quantitative growth of linear recurrences

In 1982, Schlickewei and Van der Poorten claimed that any multi-recurrence sequence has, essentially, maximal possible growth rate. Fourty years later, Fuchs and Heintze provided a non-effective proof of this statement. In this paper, we…

Let $S= \{ p_1, \ldots, p_s\}$ be a finite, non-empty set of distinct prime numbers and $(U_{n})_{n \geq 0}$ be a linear recurrence sequence of integers of order $r$. For any positive integer $k,$ we define $(U_j^{(k)})_{j\geq 1}$ an…

Let $ (G_n)_{n=0}^{\infty} $ be a non-degenerate linear recurrence sequence with power sum representation $ G_n = a_1(n) \alpha_1^n + \cdots + a_t(n) \alpha_t^n $. In this paper we will prove a function field analogue of the well known…

Generalizing some popular sequences like Catalan's number, Schr\"oder's number, etc, we consider the sequence $s_n$ with $s_0=1$ and for $n\ge 1$, \begin{multline*} s_n=\sum_{x_1+\dots+x_{\ell_1}=n-1} \kappa_1 s_{x_1}\dots s_{x_{\ell_1}} +…

Let $\{U(m)\}_{m\in \N}$ and $\{V(n)\}_{n\in \N}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set of natural numbers $n$ such that the ratio $U(n)/V(n)$ is an integer. We study…

Let $(U_n)_{n\geq 0}$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $\ell$, let $\ell \cdot 2^{\ell} + 1$ be a Cullen number. Recently in \cite{bmt}, generalized Cullen numbers in…

Let $\{ {U_{n}\}_{n \geq 0} }$ be a non-degenerate binary recurrence sequence with positive discriminant. Let $\{p_1,\ldots, p_s\}$ be fixed prime numbers and $\{b_1,\ldots ,b_s\}$ be fixed non-negative integers. In this paper, we obtain…

The orthorecursive expansion of unity with respect to the system $\{x, x^2, x^3, \ldots\}$ in $L^2([0,1])$ produces a sequence of rational coefficients $(c_n)$ defined by an explicit recurrence. Kalmynin and Kosenko established the bounds…

Let $ G_n $ and $ H_m $ be two non-degenerate linear recurrence sequences defined over a function field $ F $ in one variable over $ \mathbb{C} $, and let $ \mu $ be a valuation on $ F $. We prove that under suitable conditions there are…

Let $(u_n)_{n \geq 0}$ be a nondegenerate linear recurrence of integers, and let $\mathcal{A}$ be the set of positive integers $n$ such that $u_n$ and $n$ are relatively prime. We prove that $\mathcal{A}$ has an asymptotic density, and that…

In this paper we provide a complete proof for a bound on the growth of multi-recurrences which are defined over a number field. The proven bound was already stated by van der Poorten and Schlickewei forty years ago.

In this paper, we consider the Diophantine equation $\lambda_1U_{n_1}+\ldots+\lambda_kU_{n_k}=wp_1^{z_1} \cdots p_s^{z_s},$ where $\{U_n\}_{n\geq 0}$ is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2;…

Let $p$ and $q$ be two distinct fixed prime numbers and $(n_i)_{i\geq 0}$ the sequence of consecutive integers of the form $p^a\cdot q^b$ with $a,b\ge 0$. Tijdeman gave a lower bound (1973) and an upper bound (1974) for the gap size…

Let $(U_n)_{n=0}^\infty$ and $(V_m)_{m=0}^\infty$ be two linear recurrence sequences. For fixed positive integers $k$ and $\ell$, fixed $k$-tuple $(a_1,\dots,a_k)\in \mathbb{Z}^k$ and fixed $\ell$-tuple $(b_1,\dots,b_\ell)\in…

We describe an algorithm that takes as input a complex sequence $(u_n)$ given by a linear recurrence relation with polynomial coefficients along with initial values, and outputs a simple explicit upper bound $(v_n)$ such that $|u_n| \leq…

W.M.Schmit[11] conjectured that for any$\;\theta$ with deg$\;\theta\geq 3,$ there is no constant$\;C=C(\theta)$ so that$\;|p-q\theta|>Cq^{-1}$ for every rationa$\;p/q.$ [12,p26] states that the computations of the first several thousand…

We consider the recursion $X_{n+1}=\sum_{i=0}^n \epsilon_{n,i}X_{n-i}$, where $\epsilon_{n,i}$ are i.i.d. (Bernoulli) random variables taking values in $\{-1,1\}$, and $X_0=1$, $X_{-j}=0$ for $j>0$. We prove that almost surely, $n^{-1}\log…

Let $(u_n)_{n \geq 0}$ be a non-degenerate Lucas sequence, given by the relation $u_n=a_1 u_{n-1}+a_2 u_{n-2}$. Let $\ell_u(m)=lcm(m, z_u(m))$, for $(m,a_2)=1$, where $z_u(m)$ is the rank of appearance of $m$ in $u_n$. We prove that…

Let $(a_n), (b_n)$ be linear recursive sequences of integers with characteristic polynomials $A(X),B(X)\in \mathbb{Z}[X]$ respectively. Assume that $A(X)$ has a dominating and simple real root $\alpha$, while $B(X)$ has a pair of conjugate…

Let $S = \{q_1, \ldots , q_s\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m = q_1^{r_1} \ldots q_s^{r_s} M$, where $r_1, \ldots , r_s$ are non-negative integers and $M$ is an integer relatively…