Related papers: Linear recurrences indexed by $\mathbb{Z}$
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…
For enumerative problems, i.e. computable functions f from N to Z, we define the notion of an effective (or closed) formula. It is an algorithm computing f(n) in the number of steps that is polynomial in the combined size of the input n and…
We provide a simple method to recognize classical orthogonal polynomials on lattices defined only by their coefficients of the three term recurrence relation.
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
We consider the problem of deciding $\omega$-regular properties on infinite traces produced by linear loops. Here we think of a given loop as producing a single infinite trace that encodes information about the signs of program variables at…
Let $ K $ be a number field, $ S $ a finite set of places of $ K $, and $ \mathcal{O}_S $ be the ring of $ S $-integers. Moreover, let $$ G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G_n^{(d)} $$ be a polynomial in $ Z $ having simple linear…
Linear constraints are the linear counterpart of Haskell's class constraints. Linearly typed parameters allow the programmer to control resources such as file handles and manually managed memory as linear arguments. Indeed, a linear type…
Lonesum matrices are matrices that are uniquely reconstructible from their row and column sum vectors. These matrices are enumerated by the poly-Bernoulli numbers that are related to the multiple zeta values and have a rich literature in…
For linear recurrence systems, the problem of finding rational solutions is reduced to the problem of computing polynomial solutions by computing a content bound or a denominator bound. There are several bounds in the literature. The…
Interior point methods (IPMs) are a common approach for solving linear programs (LPs) with strong theoretical guarantees and solid empirical performance. The time complexity of these methods is dominated by the cost of solving a linear…
One of the grand challenges of Mathematics instruction is to provide students with problems that are both accessible and have a reasonably elegant solution. Instructors commonly resort to resources like course textbooks, online-learning…
Some identities that involve the elliptic version of the Cauchy matrices are presented and proved. They include the determinant formula, the formula for the inverse matrix, the matrix product identity and the factorization formula.
We consider a heat equation and a wave equation in a spatial interval over a time interval. This article deals with inverse problems of determining sizes of spatial intervals by extra boundary data of solutions of the governing equations.…
We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are…
We briefly review the results of our paper hep-th/0009013: we study certain perturbative solutions of left-unilateral matrix equations. These are algebraic equations where the coefficients and the unknown are square matrices of the same…
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…
In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…
A systematic framework is presented for the construction of hierarchies of soliton equations. This is realised by considering scalar linear integral equations and their representations in terms of infinite matrices, which give rise to all…
We consider properties of binomial series $\sum_{n=0}^\infty a_n z^{\underline{n}}$, where $z^{\underline{n}}=z(z-1)\cdots(z-n+1)$ and the convergence of binomial series in the complex domain. The order of growth of entire and meromorphic…