On The Dynamics Of Solutions Of A Rational Difference Equation Via Generalized Tribonacci Numbers

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

arxiv: 1906.11629 · v1 · pith:IAISIX2Pnew · submitted 2019-06-27 · 🧮 math.DS

On The Dynamics Of Solutions Of A Rational Difference Equation Via Generalized Tribonacci Numbers

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

Pith reviewed 2026-05-25 14:32 UTC · model grok-4.3

classification 🧮 math.DS

keywords rational difference equationgeneralized Tribonacci numbersclosed-form solutionstabilityasymptotic behaviordynamical systems

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The pith

The solutions of the rational difference equation x_{n+1} = γ / (x_n (x_{n-1} + α) + β) are expressed using generalized Tribonacci numbers.

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

This paper studies the rational difference equation with nonnegative real parameters and initial values, deriving explicit forms for its solutions by linking them to the generalized Tribonacci sequence. It further determines the stability character of equilibria and the asymptotic behavior of the iterates. A reader would care because the association supplies closed-form expressions that replace iterative computation and directly reveal convergence or boundedness properties. The work applies this link to obtain concrete formulas and stability criteria without solving the nonlinear recurrence directly.

Core claim

The solutions of x_{n+1} = γ / (x_n (x_{n-1} + α) + β) are associated with generalized Tribonacci numbers, so that each term x_n admits an explicit representation in terms of those numbers; the same association yields the stability properties and asymptotic limits of the sequence.

What carries the argument

Generalized Tribonacci numbers, defined by a three-term linear recurrence, which furnish the closed-form expressions for the nonlinear rational iterates.

If this is right

Every solution admits a closed-form expression that can be evaluated directly for any index n.
Equilibrium stability is decided by the growth rate of the associated generalized Tribonacci sequence.
Asymptotic limits, when they exist, are determined by the limiting ratio of consecutive Tribonacci terms.
Boundedness or eventual periodicity of solutions follows from corresponding properties of the Tribonacci sequence.

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 be tested on other rational recurrences of similar degree to see whether a linear sequence of higher order emerges.
Parameter regions where the Tribonacci representation remains positive might delineate invariant domains for the map.
The explicit form allows direct verification of global attractivity without linearization at each equilibrium.

Load-bearing premise

The recurrence relation and initial conditions permit an exact closed-form representation in terms of the generalized Tribonacci sequence without additional restrictions or case distinctions that would invalidate the association for general nonnegative parameters.

What would settle it

Compute the first several terms of the difference equation for chosen nonnegative α, β, γ and initials, then compare them against the explicit formula built from the corresponding generalized Tribonacci sequence; mismatch on any term disproves the claimed association.

read the original abstract

In this study, we investigate the form of solutions, stability character and asymptotic behavior of the following rational difference equation x_{n+1}=({\gamma}/(x_{n}(x_{n-1}+{\alpha})+\b{eta})), n=0,1,..., where the inital values x_{-1} and x_{0} and {\alpha}, \b{eta} and {\gamma} with {\gamma} are nonnegative real numbers. Its solutions are associated with generalized 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 claims an explicit link between solutions of this rational recurrence and generalized Tribonacci numbers, but the association likely requires parameter restrictions that the abstract does not mention.

read the letter

The central point is that the authors give an explicit representation of the solutions to x_{n+1} = γ / (x_n (x_{n-1} + α) + β) in terms of generalized Tribonacci numbers, then use that form to study stability and asymptotics for nonnegative parameters and initial values. That is the new piece: a direct tie to this particular three-term linear sequence for this specific nonlinear equation. Similar reductions have been done for other rational recurrences, so the work is an incremental application rather than a new method. If the derivation is correct and holds for general nonnegative α, β, γ, it supplies a concrete closed form that makes stability analysis straightforward. The paper does that part cleanly enough on its own terms. The soft spot is exactly the one flagged in the stress-test note. Reducing a quadratic denominator to a linear recurrence usually forces an algebraic relation among the parameters or special initial conditions so that unwanted terms cancel. The abstract states the result for arbitrary nonnegative reals without mentioning any such restrictions or case splits. If the full paper does not supply those conditions or prove the form satisfies the original equation without them, the claim is narrower than presented. The soundness score in the report is low for the same reason: no derivation steps are visible here. This is a narrow technical note aimed at people who already work on closed-form solutions for rational difference equations. Those readers might extract a usable formula or stability criterion if the details check out. It is not broad enough to interest a general dynamics audience. I would send it to peer review so that referees can verify whether the Tribonacci representation is valid without hidden restrictions and whether the stability conclusions follow for the stated parameter range.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates the rational difference equation x_{n+1} = γ / (x_n (x_{n-1} + α) + β) for nonnegative real parameters α, β, γ and initial conditions x_{-1}, x_0. It claims that the solutions are associated with generalized Tribonacci numbers and examines their stability character and asymptotic behavior.

Significance. If the claimed association with generalized Tribonacci numbers holds exactly and without unstated restrictions on the parameters, the result would supply an explicit closed-form representation for the iterates of this nonlinear recurrence. This would be a meaningful contribution to the study of rational difference equations, as such closed forms are rare and would directly enable the stability and asymptotic analyses described.

major comments (1)

[Abstract] Abstract: the central claim that solutions are associated with generalized Tribonacci numbers for arbitrary nonnegative parameters is not accompanied by any derivation, substitution, or verification that the given nonlinear recurrence reduces to the linear Tribonacci recurrence. This is load-bearing for the result, as the association typically requires an invariant or algebraic relation among α, β, γ that may not hold in general.

minor comments (2)

[Abstract] The abstract contains a repeated phrase (γ appears twice in the parameter list) and a likely LaTeX error (``{eta}'' instead of β).
[Abstract] Typo: ``inital'' should be ``initial''.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the claimed association. We address the single major comment below and will incorporate additional clarification in the revised version.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that solutions are associated with generalized Tribonacci numbers for arbitrary nonnegative parameters is not accompanied by any derivation, substitution, or verification that the given nonlinear recurrence reduces to the linear Tribonacci recurrence. This is load-bearing for the result, as the association typically requires an invariant or algebraic relation among α, β, γ that may not hold in general.

Authors: We agree that the abstract is too concise to contain the derivation. The body of the manuscript (Section 2) defines the generalized Tribonacci sequence T_n and derives the closed-form expression for x_n by direct substitution into the given recurrence, showing that the nonlinear relation is satisfied identically for arbitrary nonnegative α, β, γ. No auxiliary algebraic constraint among the parameters is imposed or required; the invariant arises naturally from the form of the denominator. To make this load-bearing step more transparent, we will add an explicit one-paragraph outline of the substitution immediately after the statement of the main result in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation reduces solutions to known sequence without self-definition or fitted predictions

full rationale

The paper states that solutions of the given rational recurrence are associated with generalized Tribonacci numbers. No quoted step shows the association arising by construction from a fitted parameter, self-citation chain, or ansatz smuggled via prior work. The central claim is an explicit closed-form representation derived from the recurrence and initial conditions, which is independent of the target result itself. This matches the expected non-circular outcome for papers that map a nonlinear recurrence onto a linear integer sequence under stated conditions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard recursive definition of generalized Tribonacci numbers and the algebraic manipulation of the given rational recurrence; no new entities are introduced and no parameters are fitted to data.

axioms (2)

standard math Generalized Tribonacci numbers satisfy their standard three-term linear recurrence with given initial conditions.
Invoked to express the closed form of the nonlinear recurrence solutions.
domain assumption The rational recurrence admits an exact closed-form solution expressible via the Tribonacci sequence for nonnegative parameters.
This is the load-bearing assumption that allows the association stated in the abstract.

pith-pipeline@v0.9.0 · 5615 in / 1309 out tokens · 24077 ms · 2026-05-25T14:32:30.613174+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

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

By using the change of variables xn = wn-1/wn, Eq.(5) is reduced to linear third order difference equation wn+1 = β/γ wn + α/γ wn-1 + 1/γ wn-2. Set r := β/γ, s := α/γ, t := 1/γ, so we have wn+1 = r wn + s wn-1 + t wn-2.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and embed_add 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.

Its solutions are associated with generalized Tribonacci numbers... Vn+3 = r Vn+2 + s Vn+1 + t Vn

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

17 extracted references · 17 canonical work pages · 1 internal anchor

[1]

Advances in Diﬀerence Equations , 2013:174 (2013)

T ollu, DT, Y azlik, Y, T askara, N , On the solutions of two special types of Riccati dif- ference equation via Fibonacci numbers. Advances in Diﬀerence Equations , 2013:174 (2013)

work page 2013
[2]

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
[3]

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
[4]

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
[5]

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
[6]

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
[7]

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

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

work page 2017
[8]

El-Dessoky, On the dynamics of higher order diﬀerence equations xn+1 = axn + αxnxn−l βxn+γxn−k . J. Computational Analysis and Applications , 22(7): 1309-1322 (2017)

work page 2017
[9]

Mathematica Slovaca, 68(3): 625-638 (2018)

Halim, Y , Rabago, JFT , On the Solutions of a Second-Order Diﬀerence Equation in te rms of Generalized Padovan Sequences. Mathematica Slovaca, 68(3): 625-638 (2018)

work page 2018
[10]

Electronic Journal of Qualitative Theory of Diﬀerential Equations , 95: 1-18 (2018)

Stevic, S, Iricanin, B, Kosmala, W, Smarda, Z, Representation of solutions of a solvable nonlinear diﬀerence equation of second order. Electronic Journal of Qualitative Theory of Diﬀerential Equations , 95: 1-18 (2018)

work page 2018
[11]

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) ON THE DYNAMICS OF SOLUTIONS OF A RATIONAL DIFFERENCE EQUATI ON VIA GENERALIZED TRIBONACCI NUMBERS 13

work page 2018
[12]

Discrete Dynamics in Nature and Society , Article ID 9129354, 21 pages (2018)

El-Dessoky , MM, Elabbasy , EM, Asiri A, Dynamics and Solutions of a Fifth-Order Non- linear Diﬀerence Equations. Discrete Dynamics in Nature and Society , Article ID 9129354, 21 pages (2018)

work page 2018
[13]

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
[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
[15]

Asiri, A , Elsayed, EM , Dynamics and Solutions of Some Recursive Sequences of High er Order. J. Computational Analysis and Applications , 27(4): 656-670 (2019)

work page 2019
[16]

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
[17]

On Generalized Tribonacci Sedenions

Soykan Y, Okumu¸ s ˙I and T a¸ sdemir E (2019). On Generalized Tribonacci Sedenions. arXiv preprint arXiv:1901.05312

work page internal anchor Pith review Pith/arXiv arXiv 2019

[1] [1]

Advances in Diﬀerence Equations , 2013:174 (2013)

T ollu, DT, Y azlik, Y, T askara, N , On the solutions of two special types of Riccati dif- ference equation via Fibonacci numbers. Advances in Diﬀerence Equations , 2013:174 (2013)

work page 2013

[2] [2]

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

[3] [3]

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

[4] [4]

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

[5] [5]

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

[6] [6]

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

[7] [7]

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

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

work page 2017

[8] [8]

El-Dessoky, On the dynamics of higher order diﬀerence equations xn+1 = axn + αxnxn−l βxn+γxn−k . J. Computational Analysis and Applications , 22(7): 1309-1322 (2017)

work page 2017

[9] [9]

Mathematica Slovaca, 68(3): 625-638 (2018)

Halim, Y , Rabago, JFT , On the Solutions of a Second-Order Diﬀerence Equation in te rms of Generalized Padovan Sequences. Mathematica Slovaca, 68(3): 625-638 (2018)

work page 2018

[10] [10]

Electronic Journal of Qualitative Theory of Diﬀerential Equations , 95: 1-18 (2018)

Stevic, S, Iricanin, B, Kosmala, W, Smarda, Z, Representation of solutions of a solvable nonlinear diﬀerence equation of second order. Electronic Journal of Qualitative Theory of Diﬀerential Equations , 95: 1-18 (2018)

work page 2018

[11] [11]

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) ON THE DYNAMICS OF SOLUTIONS OF A RATIONAL DIFFERENCE EQUATI ON VIA GENERALIZED TRIBONACCI NUMBERS 13

work page 2018

[12] [12]

Discrete Dynamics in Nature and Society , Article ID 9129354, 21 pages (2018)

El-Dessoky , MM, Elabbasy , EM, Asiri A, Dynamics and Solutions of a Fifth-Order Non- linear Diﬀerence Equations. Discrete Dynamics in Nature and Society , Article ID 9129354, 21 pages (2018)

work page 2018

[13] [13]

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

[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

[15] [15]

Asiri, A , Elsayed, EM , Dynamics and Solutions of Some Recursive Sequences of High er Order. J. Computational Analysis and Applications , 27(4): 656-670 (2019)

work page 2019

[16] [16]

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

[17] [17]

On Generalized Tribonacci Sedenions

Soykan Y, Okumu¸ s ˙I and T a¸ sdemir E (2019). On Generalized Tribonacci Sedenions. arXiv preprint arXiv:1901.05312

work page internal anchor Pith review Pith/arXiv arXiv 2019