{"id":"6f9b2e1f-13dd-4ef4-97ea-1d647ad4f1cc","arxiv_id":"2507.03813","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sums of a polynomial times a Fibonacci-type recurrence term are shown to have one and only one representation as sequence terms at shifted indices plus a polynomial.","lead":"This paper proves that every finite sum of a polynomial times a term of a linear recurrence sequence can be written uniquely as a combination of the next few sequence terms plus a polynomial. The result generalizes classical Fibonacci sum identities to arbitrary recurrence sequences and steps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.4 is only proved for positive ℓ, yet Theorems 1.2–1.3 allow negative h; the central proof is incomplete for h<0.","rationale":"I read the construction in Section 3 and the limit argument in Section 4 in good faith. The algebra of Definitions 3.1–3.2 and Lemma 3.3 appears internally consistent, and the uniqueness argument's grouping by modulus is essentially the standard linear-independence proof for exponential polynomials; I found no counterexample. The reader's stated weakest assumption — that the Binet coefficients L_i are nonzero — is valid and follows immediately from minimal degree, so it is not the main threat. The main load-bearing gap is the negative-h case of Lemma 2.4, which is used directly in the proof of Theorem 1.2 and indirectly in Theorem 1.3. Because the gap has an evident repair and the theorem is likely true, the appropriate verdict remains conditional, matching the reader.","tokens_in":14950,"tokens_out":17446,"duration_ms":200041,"concrete_test":"Re-derive Lemma 2.4 for ℓ = −L (L > 0) by applying the positive-step argument to the reversed sequence u_k = s_{−k}, whose Binet coefficients are again L_i and whose characteristic roots are r_i^{−1}. Verify that the resulting recurrence has exactly the coefficients e_i(−L) defined in Definition 2.3. If the coefficients match, the negative-h gap is a repairable indexing slip; if they do not, compute the construction for h = −1, m = 2, r_1 = 2, r_2 = 3, P(x) = x and test identity (5) for n = 1,...,20 to see whether the negative-h case actually fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorems 1.2 and 1.3 allow h to be any nonzero integer, but the proof of the key Lemma 2.4 only covers positive ℓ. Lemma 2.4 claims that for any nonzero ℓ with r_i^ℓ distinct, s_{mℓ+q} = e_m(ℓ)s_{(m-1)ℓ+q} + ... + e_1(ℓ)s_q for all q ∈ Z. Its proof begins with an arbitrary k′ ∈ {0,1,...,ℓ−1}; for negative ℓ this index set is empty, so the residue-class argument proves nothing. This is not a cosmetic issue: Lemma 3.3 invokes (7) with ℓ = h at q = (n1+1)h+r, and Definition 3.2 constructs the polynomials P_i,d,h,r using e_i(h). Thus the existence proof for negative h rests on an unproved case of the lemma. The uniqueness proof also relies on the same step-ℓ recurrence structure when passing from the homogeneous identity to (21)–(22). The gap is probably repairable by applying the same argument to the reversed sequence s_{−k} or by using residues modulo |ℓ|, but as written the main theorem is not established over its stated domain. Separately, Section 2.1 asserts that the Binet constants L_1,...,L_m are nonzero; this is true for a minimal-degree sequence with s_1,...,s_m not all zero, but a one-line proof is omitted. The reader's conditional verdict is therefore appropriate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15153,"tokens_out":11196,"duration_ms":141846,"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":[{"comment":"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":"Lemma 2.4; Section 3, Lemma 3.3"},{"comment":"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":"Section 2.1, Eq. (6)"},{"comment":"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.","section":"Section 4, Cases ρ_w<1 and ρ_w>1"}],"minor_comments":[{"comment":"There are several typographical slips; for example, in the abstract the phrase 'h, rare integers' should read 'h, r are integers'.","section":"Abstract"},{"comment":"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":"Section 3, after Eq. (17)"},{"comment":"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.","section":"Section 4, Lemma 4.1"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The reader's conditional verdict is appropriate. The negative-h gap in Lemma 2.4 is the main correctness issue and must be fixed before the paper can be accepted. The other points—nonzero L_i and the θ=0 edge case in the uniqueness proof—are also load-bearing but admit straightforward repairs within the manuscript's scope. The paper is within the journal's scope and the central idea is sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe soft spots, in order of severity:\n\n1. 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.\n\n2. 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.\n\n3. Typos and small indexing slips (e.g., P_{m,d,h,r} vs P_{k,d,h,r} in Definition 3.2). Cosmetic.\n\nWhere 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.\n\nMy 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.","headline":"A correct and genuinely new existence/uniqueness theorem for generalized Ledin-Brousseau sums, with a fixable gap for negative h.","tokens_in":15762,"tokens_out":2942,"would_cite":false,"duration_ms":33718,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B39","11B83"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any polynomial $P$ and any distinct-root linear recurrence, the generalized Ledin-Brousseau sum has exactly one polynomial representation.","keywords":["Ledin-Brousseau sums","linear recurrence sequences","polynomial representation","uniqueness","Binet formula","finite sums","characteristic roots"],"falsifier":"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.","tokens_in":14637,"feed_emoji":"🧮","tokens_out":10214,"duration_ms":116499,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every weighted recurrence sum has one polynomial form","feed_subtitle":"For any distinct-root linear recurrence, the weighted sum has exactly one polynomial representation.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the original Fibonacci Ledin sum and its polynomial-plus-term format, which this paper extends to arbitrary distinct-root recurrences.","marker":"[5]"},{"why":"Introduces the shifted-index Brousseau sum and the finite-difference method whose polynomial-coefficient output is the pattern studied here.","marker":"[2]"},{"why":"Supplies the matrix-method derivation of recurrence relations for such sums, a direct antecedent of the step-$h$ recurrence lemma.","marker":"[9]"},{"why":"Extends the matrix method to Ledin's problem and reinforces the polynomial-representation approach used in the existence proof.","marker":"[10]"},{"why":"The author's earlier second-degree generalization is the immediate forerunner whose uniqueness question is here resolved for every degree $m$.","marker":"[4]"}],"fun_headline_variants":["Unique polynomial form for each recurrence-weighted sum","One polynomial representation per homogeneous linear sum","Existence and uniqueness for weighted recurrence sums","Recurrence sums admit exactly one polynomial form"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Unique polynomial form for each recurrence-weighted sum","One polynomial representation per homogeneous linear sum","Existence and uniqueness for weighted recurrence sums","Recurrence sums admit exactly one polynomial form"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000194,"raw_usage":{"total_tokens":1310,"prompt_tokens":856,"completion_tokens":454,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":408}},"tokens_in":472,"tokens_out":454,"duration_ms":6127,"temperature":1.0,"reasoning_tokens":408,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:03:57.147938+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ledin, On a certain kind of Fibonacci sums","cited_arxiv_id":null,"evidence_quote":"Defines the original Fibonacci Ledin sum and its polynomial-plus-term format, which this paper extends to arbitrary distinct-root recurrences."},{"cited_title":"Brousseau, Summation of Pn k=1 kmFk+r finite difference approach","cited_arxiv_id":null,"evidence_quote":"Introduces the shifted-index Brousseau sum and the finite-difference method whose polynomial-coefficient output is the pattern studied here."},{"cited_title":"Ollerton and A.G","cited_arxiv_id":null,"evidence_quote":"Supplies the matrix-method derivation of recurrence relations for such sums, a direct antecedent of the step-$h$ recurrence lemma."},{"cited_title":"Shannon and R.L","cited_arxiv_id":null,"evidence_quote":"Extends the matrix method to Ledin's problem and reinforces the polynomial-representation approach used in the existence proof."},{"cited_title":"Hadinata, On sums involving polynomials and generalized Fibonacci sequences","cited_arxiv_id":null,"evidence_quote":"The author's earlier second-degree generalization is the immediate forerunner whose uniqueness question is here resolved for every degree $m$."}],"review_version":1}