Linear Recurrence Sequences Research Articles

Series of reciprocal products with factors from linear recurrence sequences

In this article we study the values of power series ∑ z^n/\prod^j_{i = 0} W_{(n + im)k} at certain points of their domain of meromorphy from the arithmetical point of view. (W_n) is a sequence of non-zero integers satisfying a recurrence W_{n + 1} = pW_n + qW_{n − 1} with non-zero integers p, q such that the discriminant Δ = p^2 + 4q is positive but not a square. The main interest is to characterize the situations, where these values lie in the real quadratic number field ℚ(√Δ) or even in ℚ , but we also include some transcendence problems.

Quelques endomorphismes continus des suites P-récursives

Let be K a commutative field of zero characteristic, r ( K ) and j ( K ) the Hadamard algebra of linear recurrence sequences with constant coefficients respectively polynomial coefficients. In a preceding article, we have defined the map decimation ϕ d and tressage ψ d , J , σ , where d ∈ N * , J ∈ N d and σ is a permutation of { 0 , 1 , … , d − 1 } . We have shown that these maps are continuous endomorphisms of the algebra r ( K ) . In this Note, we show that these maps are also continuous endomorphisms of the algebra j ( K ) . To cite this article: A. Ait Mokhtar, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Positivity of third order linear recurrence sequences

It is shown that the Positivity Problem for a sequence satisfying a third order linear recurrence with integer coefficients, i.e., the problem whether each element of this sequence is nonnegative, is decidable.

Frequency characteristics of linear recurrence sequences over Galois rings

The frequencies of occurrences of elements in linear recurrence sequences of vectors over Galois rings are studied. The study of these frequencies is reduced to the study of the corresponding trigonometric sums over Galois rings. Based on estimates for trigonometric sums, nontrivial estimates for the frequencies of occurrence of elements in linear recurrence sequences are obtained, which generalize some known results for sequences over a finite field. These estimates are asymptotically best possible.Bibliography: 25 titles.

Studies on the distribution of the shortest linear recurring sequences

The distribution of the shortest linear recurrence (SLR) sequences in the Z/( p) field and over the Z/( p e ) ring is studied. It is found that the length of the shortest linear recurrent (SLRL) is always equal to n/2, if n is even and n/2 + 1 if n is odd in the Z/( p) field, respectively. On the other hand, over the Z/( p e ) ring, the number of sequences with length n can also be calculated. The recurring distribution regulation of the shortest linear recurring sequences is also found. To solve the problem of calculating the SLRL, a new simple representation of the Berlekamp–Massey algorithm is developed as well.

Quasi-locally finite polynomial endomorphisms

If F is a polynomial endomorphism of \({\mathbb {C}^N}\), let \({\mathbb {C} (X)^F}\) denote the field of rational functions \({r \in \mathbb C (x_1,\ldots,x_N)}\) such that \({r \circ F=r}\). We will say that F is quasi-locally finite if there exists a nonzero \({p \in \mathbb C (X)^F[T]}\) such that p(F) = 0. This terminology comes out from the fact that this definition is less restrictive than the one of locally finite endomorphisms made in Furter, Maubach (J Pure Appl Algebra 211(2):445–458, 2007). Indeed, F is called locally finite if there exists a nonzero \({p \in \mathbb C [T]}\) such that p(F) = 0. In the present paper, we show that F is quasi-locally finite if and only if for each \({a \in \mathbb C^N}\) the sequence \({n \mapsto F^n(a)}\) is a linear recurrent sequence. Therefore, this notion is in some sense natural. We also give a few basic results on such endomorphisms. For example: they satisfy the Jacobian conjecture.

Occurrence indices of elements in linear recurrence sequences over primary residue rings

We study distances to the first occurrence (occurrence indices) of a given element in a linear recurrence sequence over a primary residue ring $$ \mathbb{Z}_{p^n } $$ . We give conditions on the characteristic polynomial F(x) of a linear recurrence sequence u which guarantee that all elements of the ring occur in u. For the case where F(x) is a reversible Galois polynomial over $$ \mathbb{Z}_{p^n } $$ , we give upper bounds for occurrence indices of elements in a linear recurrence sequence u. A situation where the characteristic polynomial F(x) of a linear recurrence sequence u is a trinomial of a special form over ?4 is considered separately. In this case we give tight upper bounds for occurrence indices of elements of u.

Pattern sequences in ‹ q, r›- numeration systems

Let q ≥ 2 and 0 ≤ r ≤ q − 2 be integers. In this paper, we study pattern sequences for patterns in ‹ q, r›-numeration systems through their generating functions. Our result implies that any nontrivial linear combination over ℂ of pattern sequences chosen from different ‹ q, r›-numeration systems cannot be a linear recurrence sequence. In particular, pattern sequences in different ‹ q, r›-numeration systems are linearly independent over ℂ, while within one ‹ q, r›-numeration system they can be linearly dependent ℂ.

Locally finite polynomial endomorphisms

We study polynomial endomorphisms F of CN which are locally finite in the following sense: the vector space generated by r∘Fn (n≥0) is finite dimensional for each r∈C[x1,…,xN]. We show that such endomorphisms exhibit similar features to linear endomorphisms: they satisfy the Jacobian Conjecture, have vanishing polynomials, admit suitably defined minimal and characteristic polynomials, and the invertible ones admit a Dunford decomposition into “semisimple” and “unipotent” constituents. We also explain a relationship with linear recurrent sequences and derivations. Finally, we give particular attention to the special cases where F is nilpotent and where N=2.

On the multiplicity of linear recurrence sequences

We prove a lemma regarding the linear independence of certain vectors and use it to improve on a bound due to Schmidt on the zero-multiplicity of linear recurrence sequences.

A Skolem–Mahler–Lech theorem in positive characteristic and finite automata

Lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions. This result is known as the Skolem–Mahler–Lech theorem. Lech gave a counterexample to a similar statement in positive characteristic. We will present some more pathological examples. We will state and prove a correct analog of the Skolem–Mahler–Lech theorem in positive characteristic. The zeroes of a recurrence sequence in positive characteristic can be described using finite automata.

On the positivity set of a linear recurrence sequence

We consider real sequences (f n ) that satisfy a linear recurrence with constant coefficients. We show that the density of the positivity set of such a sequence always exists. In the special case where the sequence has no positive dominating characteristic root, we establish that the density is positive. Furthermore, we determine the values that can occur as density of such a positivity set, both for the special case just mentioned and in general.

Arithmetic functions with linear recurrence sequences

We obtain asymptotic formulas for all the moments of certain arithmetic functions with linear recurrence sequences. We also apply our results to obtain asymptotic formulas for some mean values related to average orders of elements in finite fields.

Some divisibilities amongst the terms of linear recurrences

Let (un)n≥0 be a non-degenerate linear recurrence sequence of integers. We show that the set of positive integersn such that either ω)(n) orΩ(n) dividesun is of asymptotic density zero, where ω(n) and Ω(n) are the numbers of prime and prime power divisors ofn, respectively. The same also holds for the set of positive integersn such that τ(n)un, where τ(n) is the number of the positive integer divisors of n, provided thatun satisfies some mild technical conditions.

Low-Correlation, High-Nonlinearity Sequences for Multiple-Code CDMA

Families of binary low-correlation sequences with high nonlinearity (in relation with their Walsh Hadamard transform) are constructed by using the most significant bit of linear recurrence sequences over the ring Zopf2 l, for lges3. The engineering motivation is the design of a multiple-code code-division multiple-access (CDMA) scheme with a control of low peak-to-average power ratio (PAPR). Proof techniques combine Galois ring theory (local Weil bound) with spectral analysis over the additive group of Zopf2 l. New estimates on the size of weighted degree trace codes are derived. The parameters of the sequences families constructed are shown to lie above a modified Varshamov-Gilbert bound

Arithmetical properties of linear recurrent sequences

Let F ( z ) ∈ R [ z ] be a polynomial with positive leading coefficient, and let α > 1 be an algebraic number. For r = deg F > 0 , assuming that at least one coefficient of F lies outside the field Q ( α ) if α is a Pisot number, we prove that the difference between the largest and the smallest limit points of the sequence of fractional parts { F ( n ) α n } n = 1 , 2 , 3 , … is at least 1 / ℓ ( P r + 1 ) , where ℓ stands for the so-called reduced length of a polynomial.

On the Diophantine equation x2-dy4=1 with prime discriminant II

Let $p$ denote a prime number. P. Samuel recently solved the problem of determining all squares in the linear recurrence sequence $\{ T_n \}$, where $T_n$ and $U_n$ satisfy $T_n^2-pU_n^2=1$. Samuel left open the problem of determining all squares in

Positivity of second order linear recurrent sequences

We give a decision method for the Positivity Problem for second order recurrent sequences: it is decidable whether or not a recurrent sequence defined by u n = au n - 1 + bu n - 2 has only nonnegative terms.

Hopf Algebras of Linear Recurring Sequences over Rings and Modules

The module of linear recurring sequences over a commutative ring R can be considered as a Hopf algebra dual to the polynomial Hopf algebra over R. Under this approach, some notions and operations from the Hopf algebra theory have an interesting interpretation in terms of linear recurring sequences. Generalizations are also considered: linear recurring bisequences, sequences over modules, and k-sequences.

On the nonlinearity of linear recurrence sequences

We obtain an upper bound on exponential sums of a new type with linear recurrence sequences. We apply this bound to estimate the Fourier coefficients, and thus the nonlinearity, of a Boolean function associated with a linear recurrence sequence in a natural way.