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 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 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 Is the Sibuya distribution a progeny?

2019 ◽  
Vol 56 (01) ◽  
pp. 52-56
Author(s):  
Gérard Letac
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Generating Function ◽  
Branching Process ◽  
Power Series Expansion ◽  
Nonnegative Coefficients ◽  
Positive Integers

AbstractFor 0 &lt; a &lt; 1, the Sibuya distribution sa is concentrated on the set ℕ+ of positive integers and is defined by the generating function $$\sum\nolimits_{n = 1}^\infty s_a (n)z^{{\kern 1pt} n} = 1 - (1 - z)^a$$. A distribution q on ℕ+ is called a progeny if there exists a branching process (Zn)n≥0 such that Z0 = 1, such that $$(Z_1 ) \le 1$$, and such that q is the distribution of $$\sum\nolimits_{n = 0}^\infty Z_n$$. this paper we prove that sa is a progeny if and only if $${\textstyle{1 \over 2}} \le a &#x003C; 1$$. The main point is to find the values of b = 1/a such that the power series expansion of u(1 − (1 − u)b)−1 has nonnegative coefficients.

On the propagation functions of quantum electrodynamics

1954 ◽  
Vol 225 (1163) ◽  
pp. 535-548 ◽  
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Fourth Order ◽  
Quantum Electrodynamics ◽  
Power Series Expansion ◽  
Vertex Part ◽  
Self Energy ◽  
One To One ◽  
Renormalization Constants

Schwinger’s equations for the propagation functions of quantum electrodynamics are redefined in a way to give the finite (renormalized) propagation functions without reference to divergent integrals or infinite renormalization constants. This is achieved by incorporating in the equations themselves a limiting process which is an extension of that introduced by Dirac and Heisenberg. The formulation is given independently of the power-series expansion, but the cancellation of singularities is established only in terms of such an expansion. The method is illustrated first by considering the lowest-order approximations. The lowestorder electron self-energy and vertex-part expressions are worked out, and the compensation of the singularities corresponding to the ‘ b ’ divergences is indicated in the fourth order. In the power-series expansion, the prescriptions are in a one-to-one correspondence to those of Dyson. Their formulation independently of this expansion sums up the rules obtained in the different approximations.

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.

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