Linear Recurrence Sequences Research Articles

On Gelfond-type problem for generalized Zeckendorf representations

Gelfond proved that for coprime 𝑏 − 1 and 𝑑 sums of digits of 𝑏-ary expressions of natural numbers are uniformly distributed on arithmetic progressions with the common difference 𝑑. Later, similar result was proved for the representations of natural numbers based on linear recurrent sequences. We consider the remainder term of the corresponding asymptotic formula and study the dichotomy between the logarithmic and power estimates of the remainder term. In the case 𝑑 = 2, we obtain some sufficient condition for the validity of the logarithmic estimate. Using them, we show that the logarithmic estimate holds for expansions based on all second-order linear recurrenct sequences and on infinite family of third-order sequences. Also we construct an example of the linear reccurent sequence of an arbitrary order with such property. On the other hand, we give an example of a third-order linear recurrent sequence for which the logarithmic estimate does not hold. We also show that for 𝑑 = 3 the logarithmic estimate does not hold even in the simplest case of the expansions based on Fibonacci numbers. In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. In this case, we prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference 2 with the logarithmic remainder term.

A new class of generating functions of binary products of Gaussian numbers and polynomials

In this paper, we introduce a operator in order to derive some new symmetric properties of Gaussian Fibonacci numbers and Gaussian Lucas numbers. By making use of the operator defined in this paper, we give some new generating functions for Gaussian Fibonacci numbers and Gaussian Jacobsthal polynomials. In the paper Al4, Al5, a second-order linear recurrence sequence (U_{n}(a,b;p,q))_{n≥0} or briefly (U_{n})_{n≥0} is considered by the recurrence relation: U_{n+2}=pU_{n+1}+qU_{n}, with the initial conditions U₀=a and U₁=b, where a,b∈ℂ and p,q∈ℤ₊ for n≥0.

A function field variant of Pillai's problem

In this paper, we consider a variant of Pillai's problem over function fields F in one variable over C. For given simple linear recurrence sequences Gn and Hm, defined over F and satisfying some weak conditions, we will prove that the equation Gn−Hm=f has only finitely many solutions (n,m)∈N2 for any non-zero f∈F, which can be effectively bounded. Furthermore, we prove that under suitable assumptions there are only finitely many effectively computable f with more than one representation of the form Gn−Hm.

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.

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.

On the zero-multiplicity of the k-generalized Fibonacci sequence

The k-generalized Fibonacci sequence is the linear recurrent sequence of order k, whose first k terms are and each term afterwards is the sum of the preceding k terms. In this paper, we extend to negative indices and give an upper bound on the absolute value of its largest zero in terms of k. In particular, we find that the zero-multiplicity of the k-generalized Fibonacci sequence is exactly for all . We prove that the zero multiplicity of the k-generalized Fibonacci sequence is at least for all .

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.

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.

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.

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.

Möbius Randomness Law for Frobenius Traces of Ordinary Curves

AbstractRecently E. Bombieri and N. M. Katz (2010) demonstrated that several well-known results about the distribution of values of linear recurrence sequences lead to interesting statements for Frobenius traces of algebraic curves. Here we continue this line of study and establish the Möbius randomness law quantitatively for the normalised form of Frobenius traces.

D-finiteness, rationality, and height

Motivated by a result of van der Poorten and Shparlinski for univariate power series, Bell and Chen prove that if a multivariate power series over a field of characteristic 0 0 is D-finite and its coefficients belong to a finite set, then it is a rational function. We extend and strengthen their results to certain power series whose coefficients may form an infinite set. We also prove that if the coefficients of a univariate D-finite power series “look like” the coefficients of a rational function, then the power series is rational. Our work relies on the theory of Weil heights, the Manin–Mumford theorem for tori, an application of the Subspace Theorem, and various combinatorial arguments involving heights, power series, and linear recurrence sequences.

Tropical recurrent sequences

Tropical recurrent sequences are introduced satisfying a given vector (being a tropical counterpart of classical linear recurrent sequences). We consider the case when Newton polygon of the vector has a single (bounded) edge. In this case there are periodic tropical recurrent sequences which are similar to classical linear recurrent sequences. A question is studied when there exists a non-periodic tropical recurrent sequence satisfying a given vector, and partial answers are provided to this question. Also an algorithm is designed which tests existence of non-periodic tropical recurrent sequences satisfying a given vector with integer coordinates. Finally, we introduce a tropical entropy of a vector, provide some bounds on it and extend this concept to tropical multivariable recurrent sequences.

Composite values of polynomial power sums

Let (G n (x)) n=0 ∞ be a d-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let m≥2 be a given integer. We ask for n∈ℕ such that the equation G n (x)=g∘h is satisfied for a polynomial g∈ℂ[x] with degg=m and some polynomial h∈ℂ[x] with degh>1. We prove that for all but finitely many n these decompositions can be described in “finite terms” coming from a generic decomposition parameterized by an algebraic variety. All data in this description will be shown to be effectively computable.

Cross-correlation coefficients for digit sequences of uniform linear recurrent sequences over the residue ring

Изучается класс последовательностей элементов простого поля, полученных в результате выделения старшего $p$-ичного разряда из линейных рекуррентных последовательностей над кольцом вычетов по модулю $p^n$. Приводятся абсолютные и неабсолютные оценки сверху для модулей коэффициентов кросс-корреляции рассматриваемых последовательностей. Эти результаты позволяют указать условия, при которых отрезки разрядных последовательностей, полученных из различных исходных последовательностей, также являются различными.

Some New Identities of Second Order Linear Recurrence Sequences

The main purpose of this paper is using the combinatorial method, the properties of the power series and characteristic roots to study the computational problem of the symmetric sums of a certain second-order linear recurrence sequences, and obtain some new and interesting identities. These results not only improve on some of the existing results, but are also simpler and more beautiful. Of course, these identities profoundly reveal the regularity of the second-order linear recursive sequence, which can greatly facilitate the calculation of the symmetric sums of the sequences in practice.

General Linear Recurrence Sequences and Their Convolution Formulas

We extend a technique recently introduced by Chen Zhuoyu and Qi Lan in order to find convolution formulas for second order linear recurrence polynomials generated by 1 1 + a t + b t 2 x . The case of generating functions containing parameters, even in the numerator is considered. Convolution formulas and general recurrence relations are derived. Many illustrative examples and a straightforward extension to the case of matrix polynomials are shown.

Scalar Multiplication via Elliptic Net using Generalized Equivalent Sequences

Chord and tangent is a classical method to calculate the elliptic curve scalar multiplication. Alternatively, the scalar multiplication can be calculated by dividing polynomials over certain finite fields and the first elliptic net scalar multiplication was implemented on a short Weierstrass curve. The net was originated from non-linear recurrence sequences, namely as elliptic divisibility sequence. It is well known that the linear recurrence sequences have been applied in the cryptosystem as a cipher in the encryption and decryption process. From the perspective of cryptographic application, the elliptic divisibility sequence is used generally for integer factorization, solving elliptic curve discrete logarithm problem and computation of pairing or scalar multiplication. But there is a lack of contribution of these non-linear recurrence sequences in scalar multiplication. Therefore, this paper aims to discuss a generalization of the equivalent sequence of elliptic divisibility for computing scalar multiplication. The experimental results of scalar multiplication via the net and its coding in computer programming are presented. The future direction of scalar multiplication via the elliptic net is also discussed.

Bounds on the frequencies of tuples on parts of the period of linear recurring sequences over Galois rings

Abstract We study the frequencies of tuples in linear recurring sequences (LRS) of vectors over Galois rings. By means of an estimate of an exponential sum some nontrivial bounds on the frequencies of elements in LRS are derived. It is shown that these bounds are in some cases sharper than known results.

A finite analogue of the ring of algebraic numbers

We construct an analogue of the ring of algebraic numbers, living in a quotient of the product of all finite fields of prime order. We use this ring to deduce some results about linear recurrent sequences.