Linear Recurrence Research Articles

On Homogeneous Combinations of Linear Recurrence Sequences

Let (Fn)n≥0 be the Fibonacci sequence given by Fn+2=Fn+1+Fn, for n≥0, where F0=0 and F1=1. There are several interesting identities involving this sequence such as Fn2+Fn+12=F2n+1, for all n≥0. In 2012, Chaves, Marques and Togbé proved that if (Gm)m is a linear recurrence sequence (under weak assumptions) and Gn+1s+⋯+Gn+ℓs∈(Gm)m, for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ł and the parameters of (Gm)m. In this paper, we shall prove that if P(x1,…,xℓ) is an integer homogeneous s-degree polynomial (under weak hypotheses) and if P(Gn+1,…,Gn+ℓ)∈(Gm)m for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ℓ, the parameters of (Gm)m and the coefficients of P.

On Some Multinomial Sums Related to the Fibonacci Type Numbers

Abstract In this paper we investigate Fibonacci type sequences defined by kth order linear recurrence. Based on their companion matrix and its graph interpretation we determine multinomial and binomial formulas for these sequences. Moreover we present a graphical rule for calculating the words of these sequences from the Pascal’s triangle.

Geometric Aspects of Lucas Sequences, II

This article is a sequel of $\langle$Geometric aspects of Lucas sequences, I$\rangle$, which presents a way of viewing Lucas sequences in the framework of group scheme theory. This enables us to treat the Lucas sequences from a geometric and functorial viewpoint, which was suggested by Laxton $\langle$On groups of linear recurrences, I$\rangle$ and by Aoki-Sakai $\langle$Mod $p$ equivalence classes of linear recurrence sequences of degree 2$\rangle$. It is crucial in this work to employ the notion of quotients though at an elementary level.

Horadam Sequences and Tridiagonal Determinants

We consider a family of particular tridiagonal matrix determinants which can represent the general second-order linear recurrence sequences. These determinants can be changed to symmetric or skew-symmetric tridiagonal determinants. To evaluate the complex factorizations of any Horadam sequence, we evaluate the eigenvalues of some special tridiagonal matrices and their corresponding eigenvectors. We also use these determinant representations to obtain some formulas in these sequences.

Combinatorialization of Sury and McLaughlin identities and general linear recurrences by a unified approach

Combinatorialization of Sury and McLaughlin identities and general linear recurrences by a unified approach

ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

AbstractLet $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

Recurrence relations and Benford’s law

There are now many theoretical explanations for why Benford’s law of digit bias surfaces in so many diverse fields and data sets. After briefly reviewing some of these, we discuss in detail recurrence relations. As these are discrete analogues of differential equations and model a variety of real world phenomena, they provide an important source of systems to test for Benfordness. Previous work showed that fixed depth recurrences with constant coefficients are Benford modulo some technical assumptions which are usually met; we briefly review that theory and then prove some new results extending to the case of linear recurrence relations with non-constant coefficients. We prove that, for certain families of functions f and g, a sequence generated by a recurrence relation of the form $$a_{n+1} = f(n)a_n + g(n)a_{n-1}$$ is Benford for all initial values. The proof proceeds by parameterizing the coefficients to obtain a recurrence relation of lower degree, and then converting to a new parameter space. From there we show that for suitable choices of f and g where f(n) is nondecreasing and $$g(n)/f(n)^2 \rightarrow 0$$ as $$n \rightarrow \infty $$ , the main term dominates and the behavior is equivalent to equidistribution problems previously studied. We also describe the results of generalizing further to higher-degree recurrence relations and multiplicative recurrence relations with non-constant coefficients, as well as the important case when f and g are values of random variables.

Unresectable Chondrosarcomas Treated With Carbon Ion Radiotherapy: Relationship Between Dose-averaged Linear Energy Transfer and Local Recurrence.

The local control rate of chondrosarcomas treated with carbon-ion radiotherapy (CIRT) worsens as tumour size increases, possibly because of the intra-tumoural linear energy transfer (LET) distribution. This study aimed to evaluate the relationship between local recurrence and intra-tumoural LET distribution in chondrosarcomas treated with CIRT. Thirty patients treated with CIRT for grade 2 chondrosarcoma were included. Dose-averaged LET (LETd) distribution was calculated by the treatment planning system, and the relationship between LETd distribution in the planning tumour volume (PTV) and local control was evaluated. The mean LETd value in PTV was similar between cases with and without recurrence. Recurrence was not observed in cases where the effective minimum LETd value exceeded 40 keV/μm. LETd distribution in PTV is associated with local control in chondrosarcomas and patients treated with ion beams of higher LETd may have an improved local control rate for unresectable chondrosarcomas.

Linear relations for Laurent polynomials and lattice equations

A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. Recurrences with this property appear in diverse areas of mathematics and physics, ranging from Lie theory and supersymmetric gauge theories to Teichmüller theory and dimer models. In many cases where such recurrences appear, there is a common structural thread running between these different areas, in the form of Fomin and Zelevinsky’s theory of cluster algebras. Laurent phenomenon algebras, as defined by Lam and Pylyavskyy, are an extension of cluster algebras, and share with them the feature that all the generators of the algebra are Laurent polynomials in any initial set of generators (seed). Here we consider a family of nonlinear recurrences with the Laurent property, referred to as ‘Little Pi’, which was derived by Alman et al via a construction of periodic seeds in Laurent phenomenon algebras, and generalizes the Heideman–Hogan family of recurrences. Each member of the family is shown to be linearizable, in the sense that the iterates satisfy linear recurrence relations with constant coefficients. We derive the latter from linear relations with periodic coefficients, which were found recently by Kamiya et al from travelling wave reductions of a linearizable lattice equation on a six-point stencil. By making use of the periodic coefficients, we further show that the birational maps corresponding to the Little Pi family are maximally superintegrable. We also introduce another linearizable lattice equation on the same six-point stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the six-point lattice equations considered, we use the formalism of van der Kamp to construct a broad class of initial value problems with the Laurent property.

Analytical calculation of deformations of a truss for a long span covering

Introduction. The method of induction based on the number of panels is employed to derive formulas designated for deflection of a square in plan hinged rod structure, which has supports on its sides and which consists of individual pyramidal rod elements. The truss is statically determinable and symmetrical. Some features of the solution are identified on the curves constructed according to derived formulas.
 Materials and methods. The analysis of forces arising in the rods of the covering is performed symbolically using the method of joint isolation and operators of the Maple symbolic math engine. The deflection in the centre of the covering is found by the Maxwell–Mohr’s formula. The rigidity of truss rods is assumed to be the same. The analysis of a sequence of analytical calculations of trusses, having different numbers of panels, is employed to identify coefficients, designated for deflection and reaction at the supports, in the final design formula. The induction method is employed for this purpose. Common members of sequences of coefficients are derived from the solution of linear recurrence equations made using Maple operators.
 Results. Solutions, obtained for three types of loads, are polynomial in terms of the number of panels. To illustrate the solutions and their qualitative analysis, curves describing the dependence of deflection on the number of panels are made. The author identified the quadratic asymptotics of the solution with respect to the number of panels. The quadratic asymptotics is linear in height.
 Conclusions. Formulas are obtained for calculating deflection and reactions of covering supports having an arbitrary number of panels and dimensions if exposed to three types of loads. The presence of extremum points on solution curves is shown. The dependences, identified by the author, are designated both for evaluating the accuracy of numerical solutions and for solving problems of structural optimization in terms of rigidity.

ANALYTICAL CALCULATION OF DEFLECTION OF A MULTI-LATTICE TRUSS WITH AN ARBITRARY NUMBER OF PANELS

The scheme of a planar externally statically indeterminate truss with four supports is proposed. In analytical form, for several types of loads, the problem of forces in the rods and deflectionof the structure is solved, depending on the number of panels, the size and intensity of the load. The solution uses the Maple computer mathematics system. The deflectionat Midspan is determined using Maxwell – Mohr's formula, the forces in the rods – the method of cutting out nodes from the system of equilibrium equations for all nodes, which includes four reactions of the supports. By induction, a series of solutions for trusses with a consistently increasing number of panels is generalized to an arbitrary number of panels. For the elements of the sequences of coefficientare developed and are solved by homogeneous linear recurrence equations. The resulting formulas for the deflectio of the structure under various loads have the form of polynomials in the number of panels. A linear asymptotic solution for the number of panels is found. The kinematic degeneration of the structure and the distribution of node speeds corresponding to this case were found. The dependences of the reaction of supports and forces in the most compressed and stretched rods on the number of panels are determined.

On the fully anisotropic honeycomb lattice Green function, Bessel function integrals and Pearson random walks

The analytical properties of the lattice Green function for the fully anisotropic honeycomb lattice are studied, where (α 1, α 2, α 3) are anisotropy parameters with {α j ∈ (0, ∞): j = 1, 2, 3}, and w = u + iv is a complex variable in a (u, v) plane. This integral defines a single-valued analytic function G H(α 1, α 2, α 3; w) provided that a cut is made along the real axis from u = −(α 1 + α 2 + α 3) to u = (α 1 + α 2 + α 3). We show that G H(α 1, α 2, α 3; w) is a solution of a second-order linear differential equation with ten ordinary regular singular points and four apparent singular points. The apparent singularities are removed by constructing a particular differential equation of fourth order. Next, the series solution where |w| > (α 1 + α 2 + α 3), and is introduced. It is proved that, in general, satisfies a five-term linear recurrence relation. The asymptotic behaviour of as n → ∞ is also established. In order to determine the behaviour of G H(α 1, α 2, α 3; w) along the edges of the cut we define the limit function where u ∈ [−(α 1 + α 2 + α 3), (α 1 + α 2 + α 3)]. Integral representations are established for and . In particular, it is found that where J 0(z) and Y 0(z) denote Bessel functions of the first and second kind, respectively, and u ∈ (0, α 1 + α 2 + α 3). It is also demonstrated that the piecewise functions and can be sectionally evaluated exactly for all u ∈ (0, α 1 + α 2 + α 3), in terms of complete elliptic integrals of the first kind K(k), where k 2 ≡ k 2(α 1, α 2, α 3, u) is a rational function of (α 1, α 2, α 3) and u. Finally, applications of the results are made to the lattice Green function for the fully anisotropic simple cubic lattice, and to the theory of Pearson random walks in a plane. In particular, various Bessel function integrals are evaluated in order to derive a new exact formula for the mean end-to-end distance of a general three-step random walk.

Greatest common divisors with moving targets and consequences for linear recurrence sequences

We establish consequences of the moving form of Schmidt’s Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials evaluated at moving S S -unit arguments. In doing so, we complement recent work of Levin. As an additional application, we obtain results that pertain to the greatest common divisor problem for algebraic linear recurrence sequences. These observations are motivated by previous related works of Corvaja-Zannier, Levin, and others.

On the existence of telescopers for rational functions in three variables

Zeilberger's method of creative telescoping is crucial for the computer-generated proofs of combinatorial and special-function identities. Telescopers are linear differential or (q-)recurrence operators computed by algorithms for creative telescoping. Two fundamental problems related to creative telescoping are whether telescopers exist, and how to construct them efficiently when they do. In this paper, we solve the existence problem of telescopers for rational functions in three variables including 18 cases. We reduce the existence problem from the trivariate case to the bivariate case and some related problems. The existence criteria given in this paper enable us to determine the termination of algorithms for creative telescoping with trivariate rational inputs.

English

The Pell sequence $(P_n)_{n=0}^{\infty }$ is the second order linear recurrence defined by $P_n=2P_{n-1}+P_{n-2}$ with initial conditions $P_0=0$ and $P_1=1$. In this paper, we investigate a generalization of the Pell sequence called the $k$-generalized Pell sequence which is generated by a recurrence relation of a higher order. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences. Some interesting identities involving the Fibonacci and generalized Pell numbers are also deduced.

LINEAR RELATIONS AND INTEGRABILITY FOR CLUSTER ALGEBRAS FROM AFFINE QUIVERS

AbstractWe consider frieze sequences corresponding to sequences of cluster mutations for affine D- and E-type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient relations found by Keller and Scherotzke. Viewing the frieze sequence as a discrete dynamical system, we reduce it to a symplectic map on a lower dimensional space and prove Liouville integrability of the latter.

Linear recurrence sequences and the duality defect conjecture

It is conjectured that the dual variety of every smooth nonlinear subvariety of dimension $> \frac{2N}{3}$ in projective $N$-space is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of Hartshorne's complete intersection conjecture but nevertheless remains open for the case of subvarieties of codimension $> 2$. A combinatorial approach to proving the conjecture in the codimension $2$ case was developed by Holme, and following this approach Oaland devised an algorithm for proving the conjecture in the codimension $3$ case for particular $N$. This combinatorial approach gives a potential method of proving the duality defect conjecture in many of the cases by studying the positivity of certain homogeneous integer linear recurrence sequences. We give a generalization of the algorithm of Oaland to the higher codimension cases, obtaining with this bounds the degrees of counterexamples would have to satisfy, and using the relationship with recurrence sequences we prove that the conjecture holds in the codimension $3$ case when $N$ is odd.

Mersenne-Horadam identities using generating functions

The main object of the present paper is to reveal connections between Mersenne numbers $M_n=2^n-1$ and generalized Fibonacci (i.e., Horadam) numbers $w_n$ defined by a second order linear recurrence $w_n=pw_{n-1}+qw_{n-2}$, $n\geq 2$, with $w_0=a$ and $w_1=b$, where $a$, $b$, $p>0$ and $q\ne0$ are integers. This is achieved by relating the respective (ordinary and exponential) generating functions to each other. Several explicit examples involving Fibonacci, Lucas, Pell, Jacobsthal and balancing numbers are stated to highlight the results.

Progressões Aritméticas de Ordem Superior e Recorrências Lineares

Neste artigo apresentamos algumas relações entre progressões aritméticas de ordem superior e recorrências lineares com coeficientes constantes. De maneira particular, apresentamos uma nova prova usando as recorrências lineares de um resultado clássico que relaciona as progressões aritméticas de ordem superior com os polinômios. Tal prova afirma que o termo geral de uma sequência é um polinômio de grau k se, e somente se, essa sequência é uma progressão aritmética de ordem k.

Another simple proof of Graham and Pollak’s theorem

Let T be a tree with n vertices, and Dn be the distance matrix of T. Graham and Pollak (1971) discovered an elegant formula for the determinant of Dn: det(Dn)=−(n−1)(−2)n−2. It is surprising that it depends only on the order of T, not on the specific structure of T. By virtue of the classical Dodgson’s determinant-evaluation rule, Yan and Yeh (2006) presented a simple proof of the formula above. In this note, we give another simple proof, based on a homogeneous linear three-term recurrence relation: det(Dn)+4det(Dn−1)+4det(Dn−2)=0.