Existence and Uniqueness Property On a Generalized Ledin-Brousseau Sum

Ivan Hadinata

REVIEW 3 major objections 4 minor 11 references

Existence and Uniqueness Property On a Generalized Ledin-Brousseau Sum

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For any polynomial $P$ and any distinct-root linear recurrence, the generalized Ledin-Brousseau sum has exactly one polynomial representation.

desk verdict A correct and genuinely new existence/uniqueness theorem for generalized Ledin-Brousseau sums, with a fixable gap for negative h. read the letter →

arxiv 2507.03813 v2 pith:PIEKNTND submitted 2025-07-04 math.NT

Ivan Hadinata This is my paper · ORCID

classification math.NT MSC 11B3911B83

keywords Ledin-BrousseausumslinearrecurrencesequencespolynomialrepresentationuniquenessBinetformulafinitecharacteristicroots

topics The Collatz Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that a whole family of finite sums — $\sum_{k=1}^n P(k)s_{hk+r}$, where $P$ is a polynomial, $h,r$ are integers, and $(s_k)$ is a homogeneous linear recurrence of degree $m$ with distinct characteristic roots — can be rewritten, for every positive integer $n$, as $\sum_{k=1}^m P_k(n)s_{(n+k)h+r} + P_{m+1}(n)$ with polynomials $P_1,\ldots,P_{m+1}$. The new content is uniqueness: for each $P$, this $(m+1)$-tuple of polynomials is unique. So the infinite sequence of identities collapses to a canonical polynomial object attached to the recurrence, and any two representations of the same sum must coincide. This unifies the classical Fibonacci, Gibonacci, Horadam, $k$-Fibonacci and Tribonacci Ledin-Brousseau identities as special cases of one theorem.

What carries the argument

The carrier of the argument is the Binet representation $s_k=L_1r_1^k+\cdots+L_mr_m^k$, where $r_i$ are the distinct characteristic roots and $L_i$ are nonzero constants. Two limit lemmas do the heavy lifting: Lemma 2.1 says a polynomial that tends to $0$ along an infinite subset of $\mathbb{N}$ is the zero polynomial, and Lemma 2.6 says a sum of distinct unit-modulus exponentials with polynomial coefficients can converge only if every coefficient is $0$. Lemma 2.4 supplies the fixed coefficients $e_i(h)$ of the step-$h$ subsequence recurrence, and the invertible matrix in (34) is the final device that converts vanishing of the $Q_i$'s into vanishing of the $\gamma_i$'s.

What would settle it

Take $m=2$, a recurrence with two distinct roots of modulus $1$, say $s_k=e^{ik\alpha}+e^{ik\beta}$, set $h=1$, and look for nonzero polynomials $\gamma_1,\gamma_2,\gamma_3$ with $\gamma_1(n)e^{i(n+1)\alpha}+\gamma_2(n)e^{i(n+1)\beta}+\gamma_3(n)=0$ for all $n\in\mathbb{N}$. Lemma 2.6 says no such tuple exists; producing one, or finding a $P$ with two different tuples in (5) for a concrete recurrence, would falsify Theorem 1.3.

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Extended reading notes

Core claim

Theorem 1.3 states that under the hypotheses of Theorem 1.2 — $h\neq 0$, and the $h$-th powers of the characteristic roots are distinct and not equal to $1$ — the identity (5) determines exactly one tuple $(P_1,\ldots,P_{m+1})\in\mathbb{C}[x]^{m+1}$. Existence is shown by an explicit induction that builds the polynomials from an invertible matrix $M(d,h)$ and the step-$h$ recurrence coefficients $e_i(h)$. Uniqueness is shown by proving that the only solution of the homogeneous relation $\gamma_1(n)s_{(n+1)h+r}+\cdots+\gamma_m(n)s_{(n+m)h+r}+\gamma_{m+1}(n)=0$ for all $n\in\mathbb{N}$ is $\gamma_1=\cdots=\gamma_{m+1}=0$; the proof passes through the Binet form, groups roots by modulus, and uses Lemma 2.6 to rule out every nontrivial oscillatory term, ending with an invertible Vandermonde system (34) that forces each $\gamma_i$ to vanish.

Load-bearing premise

The load-bearing premise is that all $m$ coefficients $L_i$ in the exponential representation of the recurrence are nonzero, because a zero coefficient would make the final Vandermonde argument unable to force the corresponding polynomial to vanish.

Editorial extensions

If this is right

Every polynomial-weighted sum over a step-$h$ subsequence of a distinct-root recurrence has a finite polynomial certificate: the data $P_1,\ldots,P_{m+1}$ store the whole infinite family of sums.
For a fixed recurrence and step $h$, the representation is linear in $P$: once the monomial cases $P(x)=x^d$ are computed, the tuple for any polynomial is the same linear combination of these monomial tuples.
The theorem gives one uniform proof for all previously treated special sequences (Fibonacci, Gibonacci, Horadam, $k$-Fibonacci, Tribonacci), replacing case-by-case summation formulas.
The construction is algorithmic: $P_{m,d}$ comes from solving an invertible linear system with matrix $M(d,h)$, so the tuple can be computed in finitely many operations once the roots or the step-$h$ recurrence coefficients are known.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A boundary condition the paper leaves implicit: the proof needs every Binet coefficient $L_i$ to be nonzero. Minimality of the recurrence guarantees this, but if the setup were weakened to zero coefficients, the final Vandermonde step would no longer force uniqueness.
Because uniqueness is proven through Lemma 2.6, the same argument should carry over to sequences whose roots have equal moduli but distinct arguments; the hypotheses on $h$ already allow this, so no separate case is needed for 'all roots on the unit circle'.
One practical extension would be to use the canonical tuple as a normal form: precompute it for $P=x^0,\ldots,x^d$ once per recurrence, then evaluate any later weighted sum by linear combination, avoiding repeated summation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version: this is a real, niche result. It unifies a bunch of Fibonacci/Gibonacci/Horadam summation identities into one existence and uniqueness theorem for arbitrary degree-m linear recurrences with distinct characteristic roots. The uniqueness statement is new; the cited literature only does specific sequences and never asks whether the polynomial tuple is unique. The proof is self-contained and mostly standard — induction, Binet, Vandermonde, growth arguments. No fitted parameters, no empirical claims, no hidden circularity. The one self-citation is just related work, not load-bearing.

What the paper does well: the constructive part (Section 3) is explicit and verifiable. Definition 3.2 gives an algorithm for the polynomials P_i, and Lemma 3.3 checks the induction step cleanly. The uniqueness proof (Section 4) reduces to showing that any tuple vanishing on all n must be identically zero, using a three-case modulus analysis and Lemma 2.6. I went through the algebra; it works for h > 0.

The soft spots, in order of severity:

1. Lemma 2.4 is proved only for positive ℓ. The proof starts with k' ∈ {0,1,...,ℓ-1}; for ℓ < 0 that set is empty. Yet Theorems 1.2 and 1.3 state the result for all nonzero h, and Lemma 3.3 and the uniqueness proof both invoke (7) with ℓ = h. So as written, the main theorems are not established for negative h. This is the real gap. I think it's repairable — run the same argument on {0,...,|ℓ|-1} or use the reversed sequence — but the paper doesn't do it.

2. Section 2.1 asserts the Binet coefficients L_1,...,L_m are nonzero. That follows from minimal degree and distinct roots, but no proof is given. Minor.

3. Typos and small indexing slips (e.g., P_{m,d,h,r} vs P_{k,d,h,r} in Definition 3.2). Cosmetic.

Where does this leave the paper? It's a solid contribution to classical sequence theory, but with limited scope. No applications are given, and the generalization is more about packaging known techniques than about a new idea. Still, it is honest progress, and the uniqueness result genuinely isn't in the prior literature.

My recommendation: send it to a competent referee. The negative-h gap is exactly the kind of thing a referee can catch, and the author should be able to fix it. I would not desk-reject this. For my own work: I won't cite it in the next year, but if I worked on Fibonacci-like sums I would. Bring it to reading group? Maybe, if you want to discuss how much generality is worth the added abstraction.

Referee Report

3 major / 4 minor

Summary. The paper studies the finite sum ∑_{k=1}^n P(k)s_{hk+r}, where (s_k)_{k∈Z} is a bi-infinite homogeneous linear recurrence of minimal degree m with distinct characteristic roots, P is a polynomial over C, and h,r are integers with h≠0. The two main theorems assert, respectively, the existence (Theorem 1.2) and uniqueness (Theorem 1.3) of m+1 polynomials P_1,…,P_{m+1} such that the sum equals ∑_{k=1}^m P_k(n)s_{(n+k)h+r} plus P_{m+1}(n) for every n∈N. The proof constructs the polynomials explicitly via an invertible matrix (Definitions 3.1–3.2), proves the representation by induction (Lemma 3.3), and then proves uniqueness by showing that the homogeneous identity has only the trivial polynomial solution, using a modulus/limit argument. The core algebra of the construction and of the uniqueness proof is consistent, but the manuscript does not fully cover the stated domain h∈Z\{0}, because a key step-ℓ recurrence lemma is proved only for positive ℓ.

Significance. If the theorems are established in full, the paper provides a canonical polynomial representation for polynomial-weight sums over arithmetic progressions of a general linear recurrence sequence, thereby extending the Ledin–Brousseau results beyond Fibonacci, Horadam, and Tribonacci settings. A genuine strength is that the existence proof is explicit: the polynomials in Definition 3.2 are given by a parameter-free matrix inversion, and the proofs are self-contained with no fitted constants or hidden numerical normalization. The main technical gap—Lemma 2.4 being proved only for positive ℓ—is local and repairable, as is the omitted justification that the Binet coefficients are nonzero. The result is modest in scope but solid in structure, so its value depends on the corrections being made.

major comments (3)

[Lemma 2.4; Section 3, Lemma 3.3] Lemma 2.4 is proved only for positive ℓ, because its proof fixes k′ in {0,1,…,ℓ−1}; for negative ℓ this set is empty and the residue-class argument proves nothing. This matters directly: Theorems 1.2 and 1.3 allow every nonzero integer h, and Lemma 3.3 invokes (7) with ℓ=h and q=(n1+1)h+r, while the uniqueness proof in Section 4 uses the same step-h recurrence structure when passing from (20) to (21)–(22). The gap is repairable by defining T_t=s_{q+tℓ} for an arbitrary q∈Z, which works for negative ℓ, or by applying the positive-ℓ case to the reversed sequence s_{−k}; as written, however, the main theorems are not proved on their stated domain.
[Section 2.1, Eq. (6)] The assertion that the Binet coefficients L_1,…,L_m are nonzero is load-bearing but is not proved. In the uniqueness proof, from Q_{i_2}(x)=0 one needs L_{i_2}r_{i_2}^{r}≠0 before concluding from the Vandermonde system (34) that γ_1=⋯=γ_m=0. Nonzero L_i does follow from minimal degree m and distinct roots: if L_j=0, then s_k lies in the span of at most m−1 exponentials and the sequence satisfies a recurrence of degree at most m−1, contradicting Definition 1.1. Please add this argument explicitly.
[Section 4, Cases ρ_w<1 and ρ_w>1] In the cases ρ_w<1 and ρ_w>1, the proof eventually applies Lemma 2.6 to a limit of the form (29), but Lemma 2.6 requires all frequencies b_i to lie in (0,2π). If, for a root in the maximum-modulus group, r_i^h is a positive real number—which is allowed when ρ_w<1 or ρ_w>1—the corresponding angle is θ=0 and exp(inθ)=1, so the hypothesis of Lemma 2.6 is not met. The argument can be repaired by moving the θ=0 terms to the right side and applying the constant-coefficient version of Lemma 2.6, or by extending Lemma 2.6 to the case b_i∈[0,2π); as written, the contradiction in these cases is incomplete.

minor comments (4)

[Abstract] There are several typographical slips; for example, in the abstract the phrase 'h, rare integers' should read 'h, r are integers'.
[Section 3, after Eq. (17)] The sentence 'the ratio of polynomials … is equal to 1 : …' is opaque; writing the corresponding vector equation explicitly would make the argument easier to verify.
[Section 4, Lemma 4.1] In the injectivity proof, the notation for the second tuple is inconsistent: the last coordinate of the second tuple is written with superscript (1) instead of (2). This is harmless but should be corrected.
[References] Reference [1] is given only as an arXiv URL without a version or journal information; the citation style is otherwise inconsistent with the remaining entries.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the existence and uniqueness proofs are self-contained; the one self-citation is introductory and non-load-bearing.

full rationale

Theorems 1.2 and 1.3 are proved by explicit construction and by a limit-based uniqueness argument, not by importing the claimed result. Section 3 constructs P_{k,d,h,r}(x) from the roots of the characteristic polynomial and the coefficient vector delta defined by the invertible matrix M(d,h), then proves identity (5) by induction using recurrence relation (7). Relation (7) is derived independently in Lemma 2.4 from the Binet form (6) and Vieta's formulas, not assumed. The uniqueness proof in Section 4 takes an arbitrary homogeneous solution (gamma_1, ..., gamma_{m+1}) of (20), converts it to exponential form (23), and forces all coefficients to vanish using Lemmas 2.1, 2.2, and 2.6 together with the nonzero Vandermonde determinant at (34). No fitted constants, empirical predictions, or normalization choices are involved. The only self-citation is reference [4], cited in the introduction as related work on a generalized second-degree recurrence; it is not used in the proofs. The proof appears to contain a nontrivial gap for negative h in Lemma 2.4, since the residue classes {0, 1, ..., ell-1} are empty when ell < 0, but that is a correctness/completeness concern rather than circularity. Accordingly the circularity score is 1, reflecting only the trace of self-citation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof uses the defining properties of the recurrence, the Binet representation with nonzero coefficients, and standard analytic facts about polynomial-exponential limits. No free parameters are fitted and no new entities are introduced.

assumptions (4)

domain assumption The sequence (s_k) satisfies Definition 1.1: linear recurrence of minimal degree m with a1 not zero and characteristic polynomial with m distinct roots.
This is the class of sequences under study. It ensures the Binet form exists and has m nonzero exponential components.
standard math Every term s_k has the explicit form L_1 r_1^k + ... + L_m r_m^k with L_i nonzero.
Derived in Section 2.1 from the Vandermonde system for the initial values; used in equation (21) and throughout the uniqueness proof.
standard math The subsequence with step h satisfies the same recurrence with roots r_i^h, so the telescoping identity (7) holds for all integers q.
Lemma 2.4; used to telescope the sum. The paper proves this only for positive h even though the theorems allow h negative.
standard math Polynomial-exponential growth bounds and non-convergence of sums of distinct unit-modulus exponential phases hold as stated.
Lemmas 2.2, 2.5, and 2.6; used to force all exponential terms to vanish in the uniqueness proof. These lemmas are proven in the paper.

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Cite this review

Pith. "Pith review of Existence and Uniqueness Property On a Generalized Ledin-Brousseau Sum." pith.science (2026). https://pith.science/paper/PIEKNTND

@misc{pith2026250703813,
  author       = {Pith},
  title        = {Pith review of: Existence and Uniqueness Property On a Generalized Ledin-Brousseau Sum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PIEKNTND}},
  note         = {Machine review of arXiv:2507.03813}
}

abstract

In this paper, we present the existence and uniqueness property on a finite sum involving a polynomial and a homogeneous linear recurrence sequence. This finite sum is of the form $\sum_{k=1}^n P(k)s_{hk+r}$ where $n$ is a positive integer, $P(x)$ is a polynomial in $\mathbb C[x]$, $h$ and $r$ are some integers, and $(s_k)_{k\in\mathbb Z}$ is a homogeneous linear recurrence sequence of degree $m\geq 2$ with some constraints.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

On Ledin and Brousseau's summation problems

K. Adegoke, On Ledin and Brousseau’s summation problems , arxiv preprint https: //arxiv.org/abs/2108.04113v3 (2022)

work page Pith review arXiv 2022
[2]

Brousseau, Summation of Pn k=1 kmFk+r finite difference approach

A. Brousseau, Summation of Pn k=1 kmFk+r finite difference approach. The Fibonacci Quarterly 5(1) (1967), 91-98

work page 1967
[3]

Dresden, On the Brousseau sums Pn i=1 ipFi

G. Dresden, On the Brousseau sums Pn i=1 ipFi. Integers Vol. 22 (2022), article A105

work page 2022
[4]

Hadinata, On sums involving polynomials and generalized Fibonacci sequences

I. Hadinata, On sums involving polynomials and generalized Fibonacci sequences. Jurnal Matematika Integratif 20(2) (2024), 233-248

work page 2024
[5]

Ledin, On a certain kind of Fibonacci sums

G. Ledin, On a certain kind of Fibonacci sums . The Fibonacci Quarterly 5(1) (1967), 45-58

work page 1967
[6]

Nair, On weighted sums of Horadam numbers

P.S. Nair, On weighted sums of Horadam numbers . Advanced in Nonlinear Variational Inequalities 27(4) (2024), 472-483

work page 2024
[7]

Nair and R

P.S. Nair and R. Karunakaran, On k-Fibonacci Brousseau sums . Journal of Integer Sequences Vol. 27 (2024), article 24.6.4

work page 2024
[8]

Nair and R

P.S. Nair and R. Karunakaran, On Brousseau Sums of Tribonacci Numbers . The Fi- bonacci Quarterly, 63(1) (2025), 22–38

work page 2025

Show all 11 references

[9]

Ollerton and A.G

R.L. Ollerton and A.G. Shannon, A note on Brousseau’s summation problem . The Fibonacci Quarterly 58(5) (2020), 190-199

2020
[10]

Shannon and R.L

A.G. Shannon and R.L. Ollerton, A note on Ledin ’s summation problem. The Fibonacci Quarterly 59(1) (2021), 47-56

2021
[11]

Zeitlin, On Summation Formulas and Identities for Fibonacci Numbers

D. Zeitlin, On Summation Formulas and Identities for Fibonacci Numbers . The Fi- bonacci Quarterly, 5(1) (1967), 1–43. Ivan Hadinata Faculty of Mathematics and Natural Sciences, Gadjah Mada University, Yogyakarta, In- donesia Email : ivanhadinata2005@mail.ugm.ac.id 16

1967

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Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

On Ledin and Brousseau's summation problems

K. Adegoke, On Ledin and Brousseau’s summation problems , arxiv preprint https: //arxiv.org/abs/2108.04113v3 (2022)

work page Pith review arXiv 2022

[2] [2]

Brousseau, Summation of Pn k=1 kmFk+r finite difference approach

A. Brousseau, Summation of Pn k=1 kmFk+r finite difference approach. The Fibonacci Quarterly 5(1) (1967), 91-98

work page 1967

[3] [3]

Dresden, On the Brousseau sums Pn i=1 ipFi

G. Dresden, On the Brousseau sums Pn i=1 ipFi. Integers Vol. 22 (2022), article A105

work page 2022

[4] [4]

Hadinata, On sums involving polynomials and generalized Fibonacci sequences

I. Hadinata, On sums involving polynomials and generalized Fibonacci sequences. Jurnal Matematika Integratif 20(2) (2024), 233-248

work page 2024

[5] [5]

Ledin, On a certain kind of Fibonacci sums

G. Ledin, On a certain kind of Fibonacci sums . The Fibonacci Quarterly 5(1) (1967), 45-58

work page 1967

[6] [6]

Nair, On weighted sums of Horadam numbers

P.S. Nair, On weighted sums of Horadam numbers . Advanced in Nonlinear Variational Inequalities 27(4) (2024), 472-483

work page 2024

[7] [7]

Nair and R

P.S. Nair and R. Karunakaran, On k-Fibonacci Brousseau sums . Journal of Integer Sequences Vol. 27 (2024), article 24.6.4

work page 2024

[8] [8]

Nair and R

P.S. Nair and R. Karunakaran, On Brousseau Sums of Tribonacci Numbers . The Fi- bonacci Quarterly, 63(1) (2025), 22–38

work page 2025