Related papers: On certain rational recursive sequences of order f…

A zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to…

Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots.

We describe the sequences {x_n}_n given by the non-autonomous second order Lyness difference equations x_{n+2}=(a_n+x_{n+1})/x_n, where {a_n}_n is either a 2-periodic or a 3-periodic sequence of positive values and the initial conditions…

Given an integer linear recurrence sequence $\langle X_n \rangle_n$, the Skolem Problem asks to determine whether there is a natural number $n$ such that $X_n = 0$. Recent work by Lipton, Luca, Nieuwveld, Ouaknine, Purser, and Worrell…

We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We propose a wide class of recursive estimation procedures for the general…

We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the…

We prove that many sequences of positive numbers $(a_n)$ defined by finite linear difference equations $a_{n+k}=c_{k-1}a_{n+k-1}+...+c_0a_n$ with suitable non negative reals coefficients $c_i$ satisfy Bendford's Law on the first digit in…

We consider the numerical solution of large-scale symmetric differential matrix Riccati equations. Under certain hypotheses on the data, reduced order methods have recently arisen as a promising class of solution strategies, by forming…

Sequence of positive integers $\{x_n\}_{n\geq1}$ is called similar to $\mathbb {N}$ respectively a given property $A$ if for every $n\geq1$ the numbers $x_n$ and $n$ are in the same class of equivalence respectively $A\enskip(x_n\sim n…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a…

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…

We study a recursion that generates real sequences depending on a parameter $x$. Given a negative $x$ the growth of the sequence is very difficult to estimate due to canceling terms. We reduce the study of the recursion to a problem about a…

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive, square-free discriminant and $p$ be a fixed prime number. In this paper, we have shown the finiteness result for the solutions of the Diophantine…

By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before…

The aim of this paper is to prove the possibility of linearization of such equations by means of introduction of new variables. For $n=2$ such a procedure is well known, when new variables are components of spinors and they are widely used…

A systematic method of summing the corrections to the renormalon residue arising from higher order renormalons is discussed.

We define the generalized Golomb triangular recursion by g_{j,s,lambda}(n) = g_{j,s,lambda}(n - s - g_{j,s,lambda}(n-j)) + \lambda j. For particular choices of the initial conditions, we show that the solution of the recursion is a non-slow…

We redefine a multiplicative group structure on the set of equivalence classes of rational sequences satisfying a fixed linear recurrence of degree two, which was defined by R. R. Laxton in his paper "On groups of linear recurrences I"…

We describe new, simple, recursive methods of construction for orientable sequences over an arbitrary finite alphabet, i.e. periodic sequences in which any sub-sequence of n consecutive elements occurs at most once in a period in either…