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 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 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.

scholarly journals Decomposable polynomials in second order linear recurrence sequences

2018 ◽  
Vol 159 (3-4) ◽  
pp. 321-346 ◽  
Author(s):  
Clemens Fuchs ◽  
Christina Karolus ◽  
Dijana Kreso
Keyword(s):  
Second Order ◽  
Linear Recurrence ◽  
Order Linear ◽  
Recurrence Sequences

scholarly journals General Linear Recurrence Sequences and Their Convolution Formulas

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 132 ◽  
Author(s):  
Paolo Emilio Ricci ◽  
Pierpaolo Natalini
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Second Order ◽  
General Linear ◽  
Linear Recurrence ◽  
Order Linear ◽  
Matrix Polynomials ◽  
Recurrence Sequences ◽  
Straightforward Extension

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.

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 Mersenne-Horadam identities using generating functions

2020 ◽  
Vol 12 (1) ◽  
pp. 34-45
Author(s):  
R. Frontczak ◽  
T. Goy
Keyword(s):  
Generating Functions ◽  
Second Order ◽  
Linear Recurrence ◽  
Order Linear ◽  
Balancing Numbers ◽  
Mersenne Numbers

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.

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