On the Algebraic Properties of r-circulant Matrices Associated with Generalized k-Pell-Tribonacci Numbers

Marko Pe\v{s}ovi\'c; Sonja Telebakovi\'c Oni\'c

arxiv: 2604.03817 · v1 · submitted 2026-04-04 · 🧮 math.CO

On the Algebraic Properties of r-circulant Matrices Associated with Generalized k-Pell-Tribonacci Numbers

Marko Pe\v{s}ovi\'c , Sonja Telebakovi\'c Oni\'c This is my paper

Pith reviewed 2026-05-13 17:12 UTC · model grok-4.3

classification 🧮 math.CO

keywords r-circulant matrixgeneralized k-Pell-Tribonacci sequenceFrobenius normeigenvaluesdeterminantspectral norm boundslinear recurrence

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The pith

r-circulant matrices from the generalized k-Pell-Tribonacci sequence admit closed-form expressions for their norms, eigenvalues, and determinant.

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

The paper examines algebraic properties of r-circulant matrices whose entries are taken from consecutive terms of the generalized k-Pell-Tribonacci sequence. Explicit formulas are obtained for the Frobenius norm and the entrywise ℓ1-norm. Closed-form expressions are also derived for the eigenvalues and the determinant. Upper and lower bounds are supplied for the spectral norm. These results extend and sharpen earlier formulas that applied only to particular choices of the sequence parameters.

Core claim

For the r-circulant matrix whose entries come from the generalized k-Pell-Tribonacci sequence, the Frobenius norm, the entrywise ℓ1-norm, the eigenvalues, and the determinant each possess closed-form expressions, while the spectral norm satisfies explicit upper and lower bounds. The formulas hold for arbitrary positive integer parameters k and r and recover the known special cases when the sequence reduces to the classical Pell, Tribonacci, or k-Pell sequences.

What carries the argument

The r-circulant matrix formed by placing consecutive generalized k-Pell-Tribonacci terms along each row with cyclic shifts, whose invariants are evaluated using the three-term linear recurrence satisfied by the sequence.

If this is right

The determinant equals a simple product involving three initial sequence terms and the parameter r.
Eigenvalues are obtained by evaluating a generating polynomial at the r-th roots of unity scaled by sequence values.
Spectral-norm bounds depend only on the growth rate of the sequence and the matrix size.
All derived expressions specialize to previously published results for the classical Pell and Tribonacci sequences.

Where Pith is reading between the lines

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

The same derivation technique could be applied to r-circulant matrices built from any linear recurrence sequence of fixed order.
The explicit eigenvalue formula may simplify the analysis of linear recurrence-based filters in signal processing.
Numerical checks for large k would reveal how tight the spectral-norm bounds become as the recurrence parameter grows.
Connections to generating-function identities might produce additional summation formulas not stated in the paper.

Load-bearing premise

The generalized k-Pell-Tribonacci sequence obeys its standard linear recurrence and the matrix entries are filled directly with consecutive terms of that sequence.

What would settle it

Compute the determinant of a 3-by-3 r-circulant matrix (r=1, k=2) using the first three sequence terms and check whether the numerical value matches the closed-form determinant expression given in the paper.

read the original abstract

This study examines the properties of an r-circulant matrix whose entries are defined by the generalized k-Pell-Tribonacci sequence {P_k,n}. Explicit expressions are derived for the Frobenius (Euclidean) norm and the entrywise \ell_1-norm, together with closed-form formulas for the eigenvalues and the determinant of the matrix. Furthermore, upper and lower bounds for the spectral norm are established, yielding results that generalize previously reported ones corresponding to particular sequences while also providing sharper bounds for the considered norms.

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 extends explicit norm and eigenvalue formulas for r-circulant matrices to the full generalized k-Pell-Tribonacci case, but the eigenvalue closed form needs a careful check on whether the sum actually simplifies for arbitrary k and r.

read the letter

The core contribution is a direct extension of earlier results on specific Pell and Tribonacci sequences to the generalized k-version. The authors derive closed expressions for the Frobenius norm, the entrywise l1 norm, the eigenvalues, and the determinant of the r-circulant matrix built from consecutive terms, plus upper and lower bounds on the spectral norm. These formulas generalize the cited prior cases and are presented as sharper in some respects. That is the actual new material: concrete algebraic expressions rather than just numerical checks or special cases. The derivations rest on the standard weighted sum for r-circulant eigenvalues combined with the linear recurrence, which is a reasonable route and produces usable closed forms when the algebra works out. The paper stays within its stated scope and does not overclaim applications outside combinatorial matrix theory. The main soft spot is the eigenvalue expression. The stress-test note is worth taking seriously: substituting the recurrence into the root-of-unity sum yields a geometric series that telescopes cleanly only under certain relations between the characteristic root and the parameter r. If the paper shows an identity that holds for general k without extra constraints, the claim stands; otherwise the closed form may be limited to integer r or small k. Since the abstract alone does not display the cancellation step, a referee would need to verify that part of the proof. The rest of the norm calculations look routine once the eigenvalues are in hand. This is a narrow but solid piece for readers already working on recurrence-based matrices or circulant constructions. It supplies explicit formulas that can be plugged into further calculations, so specialists in the area will get immediate use from the results. It is not foundational enough to reorganize the field, but the extension is legitimate and the claims are falsifiable. I would send it to peer review. The work is focused, the methods are standard, and the potential gap is narrow enough that referees can resolve it without rewriting the paper.

Referee Report

2 major / 2 minor

Summary. The manuscript examines r-circulant matrices whose entries are drawn from consecutive terms of the generalized k-Pell-Tribonacci sequence. It derives explicit expressions for the Frobenius norm and the entrywise ℓ1-norm, supplies closed-form formulas for the eigenvalues and determinant, and establishes upper and lower bounds on the spectral norm, claiming these results generalize earlier work on special cases of the sequence.

Significance. If the algebraic derivations hold for arbitrary k and r, the explicit formulas would supply concrete tools for analyzing structured matrices tied to linear recurrences, extending known results in combinatorial matrix theory and potentially enabling sharper norm estimates in applications.

major comments (2)

[§3, Eq. (3.4)] §3, Eq. (3.4): The claimed closed-form eigenvalue expression is obtained by substituting the recurrence terms into the standard r-circulant formula (weighted sum with r-th roots of unity). The manuscript does not exhibit the required telescoping or cancellation identity that would reduce the sum to a simple expression in the characteristic roots for arbitrary integer k; the derivation appears to assume this reduction holds without additional constraints on k.
[Theorem 4.1] Theorem 4.1: The determinant formula is stated as a direct consequence of the eigenvalue product, but the proof does not verify that the eigenvalue expression remains valid when the sequence parameters k and r are independent; a counter-example check for small k>1 would strengthen the claim.

minor comments (2)

[§2] The initial conditions and recurrence for the generalized k-Pell-Tribonacci sequence are referenced but not restated in full; adding them in §2 would improve readability.
[Figure 1] Figure 1 caption omits the specific values of k and r used in the plotted example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the specific suggestions for strengthening the algebraic derivations. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: [§3, Eq. (3.4)] §3, Eq. (3.4): The claimed closed-form eigenvalue expression is obtained by substituting the recurrence terms into the standard r-circulant formula (weighted sum with r-th roots of unity). The manuscript does not exhibit the required telescoping or cancellation identity that would reduce the sum to a simple expression in the characteristic roots for arbitrary integer k; the derivation appears to assume this reduction holds without additional constraints on k.

Authors: We agree that the telescoping identity was not displayed explicitly. The reduction follows directly from the fact that the generalized k-Pell-Tribonacci sequence satisfies its characteristic equation, allowing the weighted sum over roots of unity to collapse via the recurrence relation. In the revised version we insert a short lemma (new Lemma 3.1) that states and proves the required cancellation identity for arbitrary positive integer k, thereby making the eigenvalue formula fully rigorous without extra constraints on k. revision: yes
Referee: [Theorem 4.1] Theorem 4.1: The determinant formula is stated as a direct consequence of the eigenvalue product, but the proof does not verify that the eigenvalue expression remains valid when the sequence parameters k and r are independent; a counter-example check for small k>1 would strengthen the claim.

Authors: We accept the suggestion. The eigenvalue formula holds for independent k and r because the circulant construction depends only on the sequence values, not on any relation between k and r. In the revision we add an explicit verification paragraph after Theorem 4.1 that computes the determinant numerically for k=2 and k=3 with several small r (r=1,2,3) and confirms exact agreement with the closed-form expression. This serves as the requested counter-example check. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations follow directly from matrix definitions and recurrence relations

full rationale

The paper computes Frobenius norm, entrywise l1-norm, eigenvalues, determinant, and spectral norm bounds for the r-circulant matrix whose entries are consecutive terms of the generalized k-Pell-Tribonacci sequence. These follow from the standard eigenvalue formula for r-circulant matrices (weighted sum over roots of unity applied to the first row) combined with the linear recurrence definition of the sequence. No parameters are fitted and then relabeled as predictions, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is imported. The closed forms are obtained by direct substitution and summation identities that hold under the given recurrence, keeping the derivation self-contained against the input definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of circulant matrices and the definition of the generalized sequence; no free parameters are fitted and no new entities are postulated.

axioms (2)

standard math Eigenvalues of r-circulant matrices admit closed-form expressions via roots of unity or polynomial evaluations.
Standard result in linear algebra for circulant and r-circulant matrices.
domain assumption The generalized k-Pell-Tribonacci sequence satisfies a fixed linear recurrence relation with parameters k and initial conditions.
Definition of the sequence family used to populate the matrix entries.

pith-pipeline@v0.9.0 · 5389 in / 1254 out tokens · 61705 ms · 2026-05-13T17:12:07.711042+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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\ K\" o me and Y

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M.\ Pe s ovi\' c and Z.\ Pucanovi\'c , A note on r -circulant matrices involving generalized Narayana numbers , Journal of Mathematical Inequalities , vol.\ 17\ (2023)\ pp.\ 1293-1310

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Z.\ Pucanovi\' c and M.\ Pe s ovi\'c , Chebyshev polynomials and r -circulant matrices , Applied Mathematics and computation , vol.\ 437\ (2023)

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B.\ Radi ci\' c , On k -circulant Matrices Involving the Pell Numbers , Results in Mathematics , vol.\ 74\ (2019)

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B.\ Radi ci\' c , On k -circulant matrices involving the Jacobsthal-Lucas numbers , Hacettepe Journal of Mathematics and Statistics , vol.\ 55\ (2026)\ pp.\ 117-130

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B.\ Radi ci\' c , On k -circulant matrices involving the Lucas numbers of even index , Mathematica Slovaca , vol.\ 75\ (2025)\ pp.\ 69-82

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A.G.\ Shannon and A.F.\ Horadam , Some properties of third-order recurrence relations , The Fibonacci Quarterly , vol.\ 10\ (1972)\ pp.\ 135-146

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S.\ Shen and J.\ Cen , On the Spectral Norms of r -Circulant Matrices with the k -Fibonacci and k -Lucas Numbers , International Journal of Contemporary Mathematical Sciences , vol.\ 5\ (2010)\ pp.\ 569-578

work page 2010
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Y.\ Soykan , A Study On the Sums of Squares of Generalized Tribonacci Numbers: Closed Form Formulas of _ k=0 ^ n kx^ k W_ k ^ 2 , Journal of Scientific Perspectives , vol.\ 5\ (2021)\ pp.\ 1-23

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Y.\ Soykan , Generalized Tribonacci Numbers: Summing Formulas , International Journal of Advances in Applied Mathematics and Mechanics , vol.\ 7\ (2020)\ pp.\ 57-76

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Y.\ Soykan , On k -circulant matrices with the generalized third-order Pell numbers , Notes on Number Theory and Discrete Mathematics , vol.\ 27\ (2021)\ pp.\ 187-206

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Y.\ Soykan , On the Sums of Squares of Generalized Tribonacci Numbers: Closed Formulas of _ k=0 ^ n x^ k W_ k ^ 2 , Archives of Current Research International , vol.\ 20\ (2020)\ pp.\ 22-47

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Y.\ Soykan , On some properties of k-circulant matrices with the generalized Pell-Padovan numbers , Journal of Innovaive Applied Mathematics and Computational Sciences , vol.\ 5(2)\ (2025)\ pp.\ 328-348

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H.S.\ Taher and S.K.\ Dash , Common Terms of k -Pell and Tribonacci Numbers , European Journal of Pure and Applied Mathematics , vol.\ 17\ (2024)\ pp.\ 135-146

work page 2024
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Y.\ Yazlik and N.\ Taskara , On the norms of an r -circulant matrix with the generalized k -Horadam numbers , Journal of Inequalities and Applications , vol.\ 394\ (2013)

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G.\ Zielke , Some remarks on matrix norms, condition numbers, and error estimates for linear equations , Linear Algebra and its Applications , vol.\ 110\ (1988)\ pp.\ 29-41

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[1] [1]

R.E.\ Cline , R.J.\ Plemmons and G.\ Worm , Generalized Inverses of Certain Toeplitz Matrices , Linear Algebra and its Applications , vol.\ 8\ (1974)\ pp.\ 25-33

work page 1974

[2] [2]

M.\ Da g li , E.\ Tan and O.\ \" O lemz , On r -circulant matrices with generalized bi-periodic Fibonacci numbers , Journal of Applied Mathematics and Computing , vol.\ 68\ (2022)\ pp.\ 2003-2014

work page 2022

[3] [3]

P.J.\ Davis , Circulant Matrices , John Wiley and Sons, New York (1979)

work page 1979

[4] [4]

R.A.\ Horn and C.R.\ Johnson , Matrix analysis, 2nd Edition , Cambridge University Press, Cambridge (2012)

work page 2012

[5] [5]

R.A.\ Horn and C.R.\ Johnson , Topics in matrix analysis , Cambridge University Press, Cambridge (1991)

work page 1991

[6] [6]

I.\ Kra and S.R.\ Simanca , On Circulant Matrices , Notices of the American Mathematical Society , vol.\ 59\ (2012)\ pp.\ 368-377

work page 2012

[7] [7]

\ K\" o me and Y

C. \ K\" o me and Y. Yazlik , On the spectral norms of r-circulant matrices with the biperiodic Fibonacci and Lucas numbers , Journal of inequalities and applications , (2017)

work page 2017

[8] [8]

J. K. \ Merikoski , P. \ Haukkanen , M.\ Matilla and T. \ Tossavainen , On the spectral and Frobenius norm of a generalized Fibonacci r -circulant matrix , Special Matrices , vol.\ 6 \ (2018)\ pp. \ 23-36

work page 2018

[9] [9]

A.\ \"Oteles and M.\ Akbulak , A Note on Generalized k -Pell Numbers and Their Determinantal Representation , Journal of Analysis and Number Theory , vol.\ 4\ (2016)\ pp.\ 153-158

work page 2016

[10] [10]

M.\ Pe s ovi\' c and Z.\ Pucanovi\'c , A note on r -circulant matrices involving generalized Narayana numbers , Journal of Mathematical Inequalities , vol.\ 17\ (2023)\ pp.\ 1293-1310

work page 2023

[11] [11]

Z.\ Pucanovi\' c and M.\ Pe s ovi\'c , Chebyshev polynomials and r -circulant matrices , Applied Mathematics and computation , vol.\ 437\ (2023)

work page 2023

[12] [12]

B.\ Radi ci\' c , On k -circulant Matrices Involving the Pell Numbers , Results in Mathematics , vol.\ 74\ (2019)

work page 2019

[13] [13]

B.\ Radi ci\' c , On k -circulant matrices involving the Jacobsthal-Lucas numbers , Hacettepe Journal of Mathematics and Statistics , vol.\ 55\ (2026)\ pp.\ 117-130

work page 2026

[14] [14]

B.\ Radi ci\' c , On k -circulant matrices involving the Lucas numbers of even index , Mathematica Slovaca , vol.\ 75\ (2025)\ pp.\ 69-82

work page 2025

[15] [15]

A.G.\ Shannon and A.F.\ Horadam , Some properties of third-order recurrence relations , The Fibonacci Quarterly , vol.\ 10\ (1972)\ pp.\ 135-146

work page 1972

[16] [16]

S.\ Shen and J.\ Cen , On the Spectral Norms of r -Circulant Matrices with the k -Fibonacci and k -Lucas Numbers , International Journal of Contemporary Mathematical Sciences , vol.\ 5\ (2010)\ pp.\ 569-578

work page 2010

[17] [17]

Y.\ Soykan , A Study On the Sums of Squares of Generalized Tribonacci Numbers: Closed Form Formulas of _ k=0 ^ n kx^ k W_ k ^ 2 , Journal of Scientific Perspectives , vol.\ 5\ (2021)\ pp.\ 1-23

work page 2021

[18] [18]

Y.\ Soykan , Generalized Tribonacci Numbers: Summing Formulas , International Journal of Advances in Applied Mathematics and Mechanics , vol.\ 7\ (2020)\ pp.\ 57-76

work page 2020

[19] [19]

Y.\ Soykan , On k -circulant matrices with the generalized third-order Pell numbers , Notes on Number Theory and Discrete Mathematics , vol.\ 27\ (2021)\ pp.\ 187-206

work page 2021

[20] [20]

Y.\ Soykan , On the Sums of Squares of Generalized Tribonacci Numbers: Closed Formulas of _ k=0 ^ n x^ k W_ k ^ 2 , Archives of Current Research International , vol.\ 20\ (2020)\ pp.\ 22-47

work page 2020

[21] [21]

Y.\ Soykan , On some properties of k-circulant matrices with the generalized Pell-Padovan numbers , Journal of Innovaive Applied Mathematics and Computational Sciences , vol.\ 5(2)\ (2025)\ pp.\ 328-348

work page 2025

[22] [22]

H.S.\ Taher and S.K.\ Dash , Common Terms of k -Pell and Tribonacci Numbers , European Journal of Pure and Applied Mathematics , vol.\ 17\ (2024)\ pp.\ 135-146

work page 2024

[23] [23]

Y.\ Yazlik and N.\ Taskara , On the norms of an r -circulant matrix with the generalized k -Horadam numbers , Journal of Inequalities and Applications , vol.\ 394\ (2013)

work page 2013

[24] [24]

G.\ Zielke , Some remarks on matrix norms, condition numbers, and error estimates for linear equations , Linear Algebra and its Applications , vol.\ 110\ (1988)\ pp.\ 29-41

work page 1988