Related papers: Effective formulas for linear recurrence sequences…
We prove that for a positive integer a the integer sequence P(n) satisfying for all n, -infty<n<infty, the recurrence P(n)=a+P(n-phi(a)), phi(a) the Euler function, generates in increasing order all integers P(n) coprime to a.The finite…
We prove the existence of a ternary sequence of factor complexity $2n+1$ for any given vector of rationally independent letter frequencies. Such sequences are constructed from an infinite product of two substitutions according to a…
In this article, we consider systems of linear congruences in several variables and obtain necessary and sufficient conditions as well as explicit expressions for the number of solutions subject to certain restriction conditions. These…
We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed…
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…
In this note we show that if $(u_n)_{n\geqslant 1}$ is a simple linearly recurrent sequence of integers whose minimal recurrence of order $k$ involves only positive coefficients that has positive initial terms, then $(Mu_{n^s})_{n\geqslant…
It is well known that every positive integer can be expressed as a sum of nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy $F_n =F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1 =1$ and $F_2 =2$. In this paper, for any…
Cut-elimination theorems constitute one of the most important classes of theorems of proof theory. Since Gentzen's proof of the cut-elimination theorem for the system $\mathbf{LK}$, several other proofs have been proposed. Even though the…
We prove a lemma, which we call the Order Ideal Lemma, that can be used to demonstrate a wide array of log-concavity and log-convexity results in a combinatorial manner using order ideals in distributive lattices. We use the Order Ideal…
We present an algorithm which, given a linear recurrence operator $L$ with polynomial coefficients, $m \in \mathbb{N}\setminus\{0\}$, $a_1,a_2,\ldots,a_m \in \mathbb{N}\setminus\{0\}$ and $b_1,b_2,\ldots,b_m \in \mathbb{K}$, returns a…
We give an overview of the recursive characterisations of random matrix ensembles that are currently at the forefront of random matrix theory by way of studying two classes of ensembles using two different types of recursive schemes:…
There is increasing interest within the research community in the design and use of recursive probability models. Although there still remains concern about computational complexity costs and the fact that computing exact solutions can be…
The approach to the constructing explicit solutions of the recurrence relations for multi-loop integrals are suggested. The resulting formulas demonstrate a high efficiency, at least for 3-loop vacuum integrals case. They also produce a new…
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…
A new, coercive formulation of the Helmholtz equation was introduced in [Moiola, Spence, SIAM Rev. 2014]. In this paper we investigate $h$-version Galerkin discretisations of this formulation, and the iterative solution of the resulting…
Motivated by a recent work of Tr\"umper we consider the general Collatz word (up-down pattern) and the sequences following this pattern. The recurrences for the first and last sequence entries are given, obtained from repeated application…
We introduce a sequent calculus with a simple restriction of Lambek's product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a…
The multitime multiple recurrences are common in analysis of algorithms, computational biology, information theory, queueing theory, filters theory, statistical physics etc. The theoretical part about them is little or not known. That is…
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…
Gessel's famous Bessel determinant formula gives the generating function of the number of permutations without increasing subsequences of a given length. Ekhad and Zeilberger proposed the challenge of finding a suitable generalization for…