scholarly journals On Homogeneous Combinations of Linear Recurrence Sequences

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2152
Author(s):  
Marie Hubálovská ◽  
Štěpán Hubálovský ◽  
Eva Trojovská
Keyword(s):  
Fibonacci Sequence ◽  
Linear Recurrence ◽  
Degree Polynomial ◽  
Recurrence Sequence ◽  
Linear Recurrence Sequence ◽  
Recurrence Sequences ◽  
Positive Integers

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.

scholarly journals On prime powers in linear recurrence sequences

2021 ◽  
Author(s):  
Japhet Odjoumani ◽  
Volker Ziegler
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Prime Numbers ◽  
Linear Recurrence ◽  
Exceptional Set ◽  
Recurrence Sequence ◽  
Lucas Sequence ◽  
Finite Set ◽  
Linear Recurrence Sequence ◽  
Recurrence Sequences

AbstractIn this paper we consider the Diophantine equation $$U_n=p^x$$ U n = p x where $$U_n$$ U n is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on $$U_n$$ U n , we show that, for any p outside of an effectively computable finite set of prime numbers, there exists at most one solution (n, x) to that Diophantine equation. We compute this exceptional set for the Tribonacci sequence and for the Lucas sequence plus one.

scholarly journals Pellans sequence and its diophantine triples

2016 ◽  
Vol 100 (114) ◽  
pp. 259-269
Author(s):  
Nurettin Irmak ◽  
Murat Alp
Keyword(s):  
Fourth Order ◽  
Diophantine Equations ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Order Linear ◽  
Linear Recurrence Sequence ◽  
Diophantine Triples ◽  
Positive Integers

We introduce a novel fourth order linear recurrence sequence {Sn} using the two periodic binary recurrence. We call it ?pellans sequence? and then we solve the system ab+1=Sx, ac+1=Sy bc+1=Sz where a < b < c are positive integers. Therefore, we extend the order of recurrence sequence for this variant diophantine equations by means of pellans sequence.

scholarly journals ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

2020 ◽  
pp. 1-10
Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Linear Recurrences ◽  
Field Case ◽  
Power Sum ◽  
Linear Recurrence Sequence

Abstract Let $ (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.

Divisor Sums of Generalised Exponential Polynomials

1996 ◽  
Vol 39 (1) ◽  
pp. 35-46 ◽  
Author(s):  
G. R. Everest ◽  
I. E. Shparlinski
Keyword(s):  
Prime Ideal ◽  
Algebraic Integer ◽  
Linear Recurrence ◽  
Exponential Polynomial ◽  
Exponential Polynomials ◽  
Recurrence Sequence ◽  
Special Cases ◽  
Divisor Sums ◽  
Recurrence Sequences

AbstractA study is made of sums of reciprocal norms of integral and prime ideal divisors of algebraic integer values of a generalised exponential polynomial. This includes the important special cases of linear recurrence sequences and general sums of S-units. In the case of an integral binary recurrence sequence, similar (but stronger) results were obtained by P. Erdős, P. Kiss and C. Pomerance.

scholarly journals Identities Involving the Fourth-Order Linear Recurrence Sequence

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1476 ◽  
Author(s):  
Lan Qi ◽  
Zhuoyu Chen
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Generating Function ◽  
Fourth Order ◽  
Power Series Expansion ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Order Linear ◽  
Elementary Methods ◽  
Linear Recurrence Sequence

In this paper, we introduce the fourth-order linear recurrence sequence and its generating function and obtain the exact coefficient expression of the power series expansion using elementary methods and symmetric properties of the summation processes. At the same time, we establish some relations involving Tetranacci numbers and give some interesting identities.

scholarly journals Distribution of some quadratic linear recurrence sequences modulo 1

2014 ◽  
Vol 30 (1) ◽  
pp. 79-86
Author(s):  
ARTURAS DUBICKAS ◽  
Keyword(s):  
Limit Point ◽  
Fibonacci Sequence ◽  
Second Order ◽  
Linear Recurrence ◽  
Order Linear ◽  
Recurrence Sequences

We show that if a is an even integer then for every ξ ∈ R the smallest limit point of the sequence ||ξan||∞n=1 does not exceed |a|/(2|a| + 2) and this bound is best possible in the sense that for some ξ this constant cannot be improved. Similar (best possible) bound is also obtained for the smallest limit point of the sequence ||ξxn||∞n=1, where (xn)∞n=1 satisfies the second order linear recurrence xn = axn−1 + bxn−2 with a, b ∈ N satisfying a > b. For the Fibonacci sequence (Fn)∞n=1 our result implies that supξ∈R lim infn→∞ ||ξFn|| = 1/5, and e.g., in case when a > 3 is an odd integer, b = 1 and θ := a/2 + p a 2/4 + 1 it shows that supξ∈R lim infn→∞ ||ξθn|| = (a − 1)/2a.

scholarly journals Some Convolution Formulae Related to the Second-Order Linear Recurrence Sequence

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 788 ◽  
Author(s):  
Zhuoyu Chen ◽  
Lan Qi
Keyword(s):  
Power Series Expansion ◽  
Second Order ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Order Linear ◽  
Symmetry Properties ◽  
Elementary Methods ◽  
Linear Recurrence Sequence ◽  
Convolution Formula ◽  
The Relationship

The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f x ( t ) for x ∈ R with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials.

scholarly journals The density of primes dividing a particular non-linear recurrence sequence

2016 ◽  
pp. 1-30
Author(s):  
Alexi Block Gorman ◽  
Tyler Genao ◽  
Heesu Hwang ◽  
Noam Kantor ◽  
Sarah Parsons ◽  
...  
Keyword(s):  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Non Linear ◽  
Linear Recurrence Sequence
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