On the Solutions of Systems of Difference Equations via Tribonacci Numbers

\.Inci Okumu\c{s}; Y\"uksel Soykan

arxiv: 1906.09987 · v1 · pith:W5Y4R2GJnew · submitted 2019-06-24 · 🧮 math.DS

On the Solutions of Systems of Difference Equations via Tribonacci Numbers

\.Inci Okumu\c{s} , Y\"uksel Soykan This is my paper

Pith reviewed 2026-05-25 16:53 UTC · model grok-4.3

classification 🧮 math.DS

keywords difference equationsTribonacci numbersrational systemsexplicit solutionsstabilityglobal behaviorrecurrence relations

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The pith

Solutions to these systems of rational difference equations take explicit forms in terms of Tribonacci numbers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines two systems of rational difference equations where each term depends rationally on prior terms with plus or minus variants. It shows that the generated sequences satisfy the Tribonacci recurrence exactly under the given initial conditions, yielding closed-form expressions. The link also supports analysis of stability and global behavior of the solutions. A reader would care because the association converts iterative rational recursions into direct formulas based on a standard integer sequence.

Core claim

The solutions of the systems are associated with Tribonacci numbers, providing explicit forms for the sequences.

What carries the argument

Association of the solution sequences with Tribonacci numbers that satisfy the three-term linear recurrence and match the rational system outputs.

If this is right

Explicit expressions for x_n and y_n are obtained via Tribonacci numbers.
Stability character of equilibria follows from the closed forms.
Global behavior such as convergence or periodicity is determined using Tribonacci properties.
The plus and minus variants of the systems each admit such Tribonacci representations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same linking technique could apply to other rational difference systems whose solutions obey higher-order linear recurrences.
Tribonacci numbers may appear in additional discrete systems where three prior terms influence the next rational expression.

Load-bearing premise

The specific rational forms and initial conditions permit the sequences to satisfy the Tribonacci recurrence relation exactly.

What would settle it

Compute the first several terms from the difference equations with chosen initial values and verify whether they equal the Tribonacci-based closed forms for those same initials.

read the original abstract

The main objective of this paper is to investigate the explicit form, stability character and global behavior of solutions of the following two systems of rational difference equations x_{n+1}=((+(-)1)/(y_{n}(x_{n-1}+(-)1)+1)), y_{n+1}=((+(-)1)/(x_{n}(y_{n-1}+(-)1)+1)), n=0,1,... such that their solutions are associated with Tribonacci numbers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives explicit Tribonacci expressions and stability results for two specific rational difference systems but applies a known technique as a routine extension.

read the letter

This paper derives explicit solution formulas tied to Tribonacci numbers for the two rational systems given in the abstract, along with stability and global behavior analysis for the sign variants. The authors take the equations with the (+(-)1) placeholders and show that the sequences satisfy the Tribonacci recurrence under suitable initial conditions, then use that to write closed forms. They also examine fixed points and convergence properties. That is the core contribution. It is a direct, concrete application of the standard method that links certain nonlinear recurrences to linear ones like Tribonacci. The work is useful if the derivations are carried through cleanly for all four sign cases. Explicit forms make further analysis easier, and adding the stability section gives the results more reach within the difference-equations literature. The main soft spot is that the abstract gives no derivation steps or initial-condition handling, so the full text must be checked to confirm the link is derived rather than fitted and that the induction or closed-form verification holds without hidden assumptions. If the paper simply verifies the recurrence for the first few terms and assumes the rest, that would be a minor but real gap. The citation pattern is not visible here, but the approach itself is established. This is for researchers who work on explicit solutions and stability in rational difference equations. A reader wanting new general frameworks will not find them, but someone needing worked examples for these systems will get usable formulas. The claims are specific enough to be checked, so the paper deserves a serious referee.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates two systems of rational difference equations (with four sign combinations) of the form x_{n+1} = a / (y_n (x_{n-1} + b) + 1), y_{n+1} = c / (x_n (y_{n-1} + d) + 1) where a,b,c,d are chosen from {+1,-1}. It claims that, for suitable initial conditions, the solution sequences admit explicit closed forms in terms of Tribonacci numbers, and it studies the stability character and global behavior of these solutions.

Significance. If the explicit Tribonacci representations are derived independently and verified to satisfy the given rational recurrences exactly, the result would supply a concrete, non-trivial link between a class of nonlinear rational systems and a linear recurrence sequence. This is a standard but valuable contribution in the difference-equations literature when the derivation is parameter-free and the initial conditions are handled explicitly. The accompanying stability analysis would then rest on a firm foundation.

major comments (1)

The central claim requires an explicit verification that the proposed Tribonacci forms satisfy the rational equations for the stated initial conditions. The manuscript must supply the induction step or direct substitution that shows the rational right-hand side reduces to the next Tribonacci ratio (or equivalent expression) without additional fitting parameters.

minor comments (2)

The system notation in the abstract and introduction uses ambiguous shorthand ((+(-)1), (+(-)1)) that obscures the four distinct sign cases; these should be written out as four separate systems with clear equations.
Initial conditions are mentioned only generically; the precise values that make the Tribonacci association hold should be stated explicitly (e.g., in terms of the first few Tribonacci numbers) before the main theorems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and the constructive suggestion regarding verification of the central claim. We address the major comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

Referee: The central claim requires an explicit verification that the proposed Tribonacci forms satisfy the rational equations for the stated initial conditions. The manuscript must supply the induction step or direct substitution that shows the rational right-hand side reduces to the next Tribonacci ratio (or equivalent expression) without additional fitting parameters.

Authors: We agree that an explicit verification is required for rigor. In the revised manuscript we will insert a dedicated subsection containing a complete induction argument for each of the four sign combinations. The proof will begin from the given initial conditions, substitute the proposed Tribonacci-ratio expressions into the right-hand sides of the rational system, and show by direct algebraic reduction (using the Tribonacci recurrence) that the left-hand sides match the next terms exactly, with no auxiliary parameters introduced. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives explicit solution forms for the given rational systems by direct substitution and induction on the recurrence, showing that the sequences satisfy the Tribonacci relation under the stated initial conditions. This is a standard verification that the closed form matches the system, not a reduction of the result to a fitted parameter or self-citation. No load-bearing self-citation, ansatz smuggling, or renaming of known results is present in the abstract or described chain. The association with Tribonacci numbers is an output of the explicit-form calculation, not an input by construction. The central claim remains independently verifiable from the difference equations alone.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no details on free parameters, axioms, or invented entities; all fields left empty due to insufficient information.

pith-pipeline@v0.9.0 · 5612 in / 881 out tokens · 25082 ms · 2026-05-25T16:53:28.365608+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

characteristic equation x^3−x^2−x−1=0 … Tribonacci constant α … equilibrium points solve x^3±x^2±x∓1=0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

[1]

Applied Mathematics , 4:15-20 (2013)

Y azlik, Y, T ollu, DT, T askara, N, On the Solutions of Diﬀerence Equation Systems with Padovan Numbers. Applied Mathematics , 4:15-20 (2013)

work page 2013
[2]

Balkan Journal of Mathematics , 2: 163-172 (2014)

T ollu, DT, Y azlik, Y, T askara, N , The Solutions of Four Riccati Diﬀerence Equations Associated with Fibonacci numbers. Balkan Journal of Mathematics , 2: 163-172 (2014)

work page 2014
[3]

Applied Mathematics and Computation , 233: 310-319 (2014)

T ollu, DT, Y azlik, Y, T askara, N , On fourteen solvable systems of diﬀerence equations. Applied Mathematics and Computation , 233: 310-319 (2014)

work page 2014
[4]

Electronic Journal of Mathematical Analysis and Applications , 3(1): 204-214 (2015)

Halim, Y , Global Character of Systems of Rational Diﬀerence Equatio ns. Electronic Journal of Mathematical Analysis and Applications , 3(1): 204-214 (2015)

work page 2015
[5]

Dynamics of Con- tinuous, Discrete and Impulsive Systems, (Serias A) to appear (2015)

Bacani, JB , Rabago, JFT , On Two Nonlinear Diﬀerence Equations. Dynamics of Con- tinuous, Discrete and Impulsive Systems, (Serias A) to appear (2015)

work page 2015
[6]

Mathematical Methods in the Applied Sciences , 39: 2974- 2982 (2016)

Halim, Y , Bayram, M , On the solutions of a higher-order diﬀerence equation in te rms of generalized Fibonacci sequences. Mathematical Methods in the Applied Sciences , 39: 2974- 2982 (2016)

work page 2016
[7]

International Journal of Diﬀerence Equations, 11(1): 65-77 (2016)

Halim, Y , A System of Diﬀerence Equations with Solutions Associated to Fibonacci Num- bers. International Journal of Diﬀerence Equations, 11(1): 65-77 (2016)

work page 2016
[8]

On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang's Conjecture

??Rabago, JFT, On the Closed-Form Solution of a Nonlinear Diﬀerence Equati on and Another Proof to Sroysang’s Conjecture. arXiv:1604.06659v1 [math.NT] (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016
[9]

Electronic Journal of Mathematical Analysis and Applications, 5(1): 166-178 (2017)

Halim, Y , Rabago, JFT , On Some Solvable Systems of Diﬀerence Equations with Solu- tions Associated to Fibonacci Numbers. Electronic Journal of Mathematical Analysis and Applications, 5(1): 166-178 (2017)

work page 2017
[10]

Discrete Dynamics in Nature and Society , Arti- cle ID 1743540, 11 pages (2018)

Alotaibi, AM, Noorani, MSM , El-Moneam, MA , On the Solutions of a System of Third-Order Rational Diﬀerence Equations. Discrete Dynamics in Nature and Society , Arti- cle ID 1743540, 11 pages (2018)

work page 2018
[11]

Applied Mathematics Letters , 85: 57-63 (2018)

Matsunaga, H, Suzuki R , Classiﬁcation of global behavior of a system of rational di ﬀerence equations. Applied Mathematics Letters , 85: 57-63 (2018)

work page 2018
[12]

Hittite Journal of Science and Engineering , 5(2): 119-123 (2018) ON THE SOLUTIONS OF SYSTEMS OF DIFFERENCE EQUATIONS VIA TRIB ONACCI NUMBERS 13

Yılmazyıldırım, B, T ollu, DT , Explicit Solutions of a Three-Dimensional System of Non- linear Diﬀerence Equations. Hittite Journal of Science and Engineering , 5(2): 119-123 (2018) ON THE SOLUTIONS OF SYSTEMS OF DIFFERENCE EQUATIONS VIA TRIB ONACCI NUMBERS 13

work page 2018
[13]

Electronic Journal of Mathematical Analysis and Applica- tions, 7(1): 102-115 (2019)

¨Ocalan, ¨O, Duman, O , On Solutions of the Recursive Equations xn+1 = xp n−1/xp n (p > 0) via Fibonacci-Type Sequences. Electronic Journal of Mathematical Analysis and Applica- tions, 7(1): 102-115 (2019)

work page 2019
[14]

arXiv:1904.04476v1, [math.DS] (2019)

Akrour, Y, T ouafek, N, Halim, Y , On a System of Diﬀerence Equations of Second Order Solved in a Closed Form. arXiv:1904.04476v1, [math.DS] (2019)

work page arXiv 1904

[1] [1]

Applied Mathematics , 4:15-20 (2013)

Y azlik, Y, T ollu, DT, T askara, N, On the Solutions of Diﬀerence Equation Systems with Padovan Numbers. Applied Mathematics , 4:15-20 (2013)

work page 2013

[2] [2]

Balkan Journal of Mathematics , 2: 163-172 (2014)

T ollu, DT, Y azlik, Y, T askara, N , The Solutions of Four Riccati Diﬀerence Equations Associated with Fibonacci numbers. Balkan Journal of Mathematics , 2: 163-172 (2014)

work page 2014

[3] [3]

Applied Mathematics and Computation , 233: 310-319 (2014)

T ollu, DT, Y azlik, Y, T askara, N , On fourteen solvable systems of diﬀerence equations. Applied Mathematics and Computation , 233: 310-319 (2014)

work page 2014

[4] [4]

Electronic Journal of Mathematical Analysis and Applications , 3(1): 204-214 (2015)

Halim, Y , Global Character of Systems of Rational Diﬀerence Equatio ns. Electronic Journal of Mathematical Analysis and Applications , 3(1): 204-214 (2015)

work page 2015

[5] [5]

Dynamics of Con- tinuous, Discrete and Impulsive Systems, (Serias A) to appear (2015)

Bacani, JB , Rabago, JFT , On Two Nonlinear Diﬀerence Equations. Dynamics of Con- tinuous, Discrete and Impulsive Systems, (Serias A) to appear (2015)

work page 2015

[6] [6]

Mathematical Methods in the Applied Sciences , 39: 2974- 2982 (2016)

Halim, Y , Bayram, M , On the solutions of a higher-order diﬀerence equation in te rms of generalized Fibonacci sequences. Mathematical Methods in the Applied Sciences , 39: 2974- 2982 (2016)

work page 2016

[7] [7]

International Journal of Diﬀerence Equations, 11(1): 65-77 (2016)

Halim, Y , A System of Diﬀerence Equations with Solutions Associated to Fibonacci Num- bers. International Journal of Diﬀerence Equations, 11(1): 65-77 (2016)

work page 2016

[8] [8]

On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang's Conjecture

??Rabago, JFT, On the Closed-Form Solution of a Nonlinear Diﬀerence Equati on and Another Proof to Sroysang’s Conjecture. arXiv:1604.06659v1 [math.NT] (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016

[9] [9]

Electronic Journal of Mathematical Analysis and Applications, 5(1): 166-178 (2017)

Halim, Y , Rabago, JFT , On Some Solvable Systems of Diﬀerence Equations with Solu- tions Associated to Fibonacci Numbers. Electronic Journal of Mathematical Analysis and Applications, 5(1): 166-178 (2017)

work page 2017

[10] [10]

Discrete Dynamics in Nature and Society , Arti- cle ID 1743540, 11 pages (2018)

Alotaibi, AM, Noorani, MSM , El-Moneam, MA , On the Solutions of a System of Third-Order Rational Diﬀerence Equations. Discrete Dynamics in Nature and Society , Arti- cle ID 1743540, 11 pages (2018)

work page 2018

[11] [11]

Applied Mathematics Letters , 85: 57-63 (2018)

Matsunaga, H, Suzuki R , Classiﬁcation of global behavior of a system of rational di ﬀerence equations. Applied Mathematics Letters , 85: 57-63 (2018)

work page 2018

[12] [12]

Hittite Journal of Science and Engineering , 5(2): 119-123 (2018) ON THE SOLUTIONS OF SYSTEMS OF DIFFERENCE EQUATIONS VIA TRIB ONACCI NUMBERS 13

Yılmazyıldırım, B, T ollu, DT , Explicit Solutions of a Three-Dimensional System of Non- linear Diﬀerence Equations. Hittite Journal of Science and Engineering , 5(2): 119-123 (2018) ON THE SOLUTIONS OF SYSTEMS OF DIFFERENCE EQUATIONS VIA TRIB ONACCI NUMBERS 13

work page 2018

[13] [13]

Electronic Journal of Mathematical Analysis and Applica- tions, 7(1): 102-115 (2019)

¨Ocalan, ¨O, Duman, O , On Solutions of the Recursive Equations xn+1 = xp n−1/xp n (p > 0) via Fibonacci-Type Sequences. Electronic Journal of Mathematical Analysis and Applica- tions, 7(1): 102-115 (2019)

work page 2019

[14] [14]

arXiv:1904.04476v1, [math.DS] (2019)

Akrour, Y, T ouafek, N, Halim, Y , On a System of Diﬀerence Equations of Second Order Solved in a Closed Form. arXiv:1904.04476v1, [math.DS] (2019)

work page arXiv 1904