A study of a recursive sequence of polynomials revealing weighted Catalan Numbers

Elizabeth Mrema; Sophie Marques

REVIEW 3 major objections 5 minor 9 references

A study of a recursive sequence of polynomials revealing weighted Catalan Numbers

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that the coefficients of the iterated polynomial $p_n=p_{n-1}^2-2$ split into $n$-independent rational invariants, and that the diagonal invariants are weighted Catalan numbers counted by labeled ordered trees.

desk verdict A small new Catalan encoding of known Chebyshev invariants, hampered by a definition bug and a proof typo; worth reviewing after revision. read the letter →

arxiv 2501.13693 v1 pith:XDLCPBJB submitted 2025-01-23 math.CO math.NT

Sophie Marques , Elizabeth Mrema This is my paper · ORCID

classification math.COmath.NT MSC 12F0512E0512E1212E1011R1805C0505C2511B83

keywords recursivepolynomialsequencesfieldtheorycyclotomicextensionsminimalpolynomialsinvariantsweightedCatalannumbersVandermondemethod

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

This paper studies the polynomials $p_0(x)=x^2-2$, $p_n(x)=p_{n-1}(x)^2-2$, obtained by repeatedly applying the map $x\mapsto x^2-2$. Its main claim is that for each fixed $k$, the coefficient of $x^{2k}$ in $p_n$ (for every $n$ with $k\le 2^n$) has the form $\sum_{j=1}^k a_{j,k}2^{2jn}$, where the rational numbers $a_{j,k}$ are independent of $n$. The paper proves this through a coefficient recursion rooted in the field-theoretic fact that $p_n$ is the minimal polynomial of $\zeta_{2^{n+3}}+\zeta_{2^{n+3}}^{-1}$, and it gives a Vandermonde-based algorithm for computing the invariants. The final result identifies the diagonal invariant $a_{k,k}$ as a weighted Catalan number: a sum over labeled ordered trees of products of node weights. This matters because it connects a purely algebraic recursion to explicit finite tree combinatorics and offers a new way to generate weighted Catalan numbers.

What carries the argument

The load-bearing object is the reciprocal-variable identity $x^{2^{n+1}}p_n(x+x^{-1})=x^{2^{n+2}}+1$, obtained by identifying $p_n$ with the minimal polynomial of a primitive $2^{n+3}$-th root of unity. It converts coefficients of $p_n$ into solutions of a triangular binomial system, giving the recursive formula for $c_{n,2k}$. The second mechanism is the induction through $p_n=p_{n-1}^2-2$, which rewrites $c_{n,2k}$ as sums of products of earlier coefficients and forces the $n$-dependence to organize into powers $2^{2jn}$ with invariant coefficients $a_{j,k}$. Finally, the diagonal invariants are decoded combinatorially by labeled ordered trees $\mathcal{T}_k$: the recurrence for $a_{k,k}$ is exactly the gluing rule that builds a tree of label $k$ from two smaller trees whose root labels sum to $k$, and each node label $v$ carries weight $b_v$.

What would settle it

Take $n=3$ and symbolically expand $x^{16}p_3(x+x^{-1})$; the claimed identity says it must equal $x^{32}+1$. A second, sharper check: expand $p_5$ directly and compare its $x^6$ coefficient with the value predicted by the invariants $a_{1,3}=-\tfrac{1}{90}$, $a_{2,3}=\tfrac{1}{72}$, $a_{3,3}=-\tfrac{1}{360}$, namely $a_{1,3}2^{10}+a_{2,3}2^{20}+a_{3,3}2^{30}$. Any mismatch falsifies Theorem 2.5.

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Extended reading notes

Core claim

The central discovery is that the coefficient table of $p_n$ is rigid in $n$. Writing $p_n(x)=\sum_{k=0}^{2^n}c_{n,2k}x^{2k}$, the paper proves (Theorem 2.5) that for every $n$ with $1\le k\le 2^n$, $c_{n,2k}=\sum_{j=1}^k a_{j,k}2^{2jn}$, with $c_{n,0}=2$, where the $a_{j,k}$ are rational numbers defined recursively and independent of $n$. The proof begins from the reciprocal identity $x^{2^{n+1}}p_n(x+x^{-1})=x^{2^{n+2}}+1$, which turns coefficient extraction into an upper-triangular linear system (Theorem 2.1); then the recursion $p_n=p_{n-1}^2-2$ separates the powers of $2^{2n}$. For the diagonal, Proposition 3.5 shows $a_{k,k}=\sum_{T\in\mathcal{T}_k}\prod_{v\in V_T}b_v^{\delta_{v,T}}$, where $b_1=-1$, $b_v=2^{-2}(2^{2(v-1)}-1)^{-1}$ for $v>1$, and $\mathcal{T}_k$ is the set of labeled ordered trees whose cardinality is the $(k-1)$-st Catalan number.

Load-bearing premise

The construction rests on the identity $x^{2^{n+1}}p_n(x+x^{-1})=x^{2^{n+2}}+1$, equivalently on identifying $p_n$, after the substitution $x+x^{-1}$, with the minimal polynomial of a primitive $2^{n+3}$-th root of unity; if that identification failed, the coefficient recursion and the invariant decomposition would have no basis.

Editorial extensions

If this is right

For each fixed $k$, the sequence of coefficients $c_{n,2k}$ across all $n$ is described by the same $k$ rational invariants, so the infinite coefficient table of the whole family is governed by a finite triangular array.
The diagonal invariant $a_{k,k}$ can be computed by enumerating labeled ordered trees with product weights, giving a purely combinatorial formula for this family of weighted Catalan numbers.
Proposition 2.8 computes all invariants $a_{j,k}$ from finitely many coefficient values via a Vandermonde system, so only a small number of explicit expansions is needed to know every later coefficient with that $k$.
Since $p_n$ is the minimal polynomial of the cosine of a 2-power angle, the coefficient identities give a field-theoretic template for computing coefficients of other polynomial recursions that arise from minimal-polynomial substitutions.

Reading between the lines

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

The paper leaves the off-diagonal invariants $a_{j,k}$ without a combinatorial interpretation; a natural test is whether they are weighted counts of forests obtained by marking a distinguished branch in the trees of $\mathcal{T}_k$.
Recasting the proof around the reciprocal-variable identity alone, without the cyclotomic-field language, would likely extend the invariant decomposition to iterations of $x^2-c$ for other constants $c$.
The node weights $b_v=2^{-2}(2^{2(v-1)}-1)^{-1}$ have denominators built from Mersenne-type factors, so one could check whether $a_{k,k}$'s denominator or 2-adic valuation follows a simple pattern as $k$ grows, which the paper does not address.

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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

Quick take: the paper derives a weighted-Catalan tree formula for the diagonal invariants of p_n(x) = p_{n-1}(x)^2 - 2, p_0 = x^2-2. The formula is new, but the paper never mentions that p_n is 2 T_{2^{n+1}}(x/2), which makes most of the coefficient results classical.

What is good: Proposition 3.5 expresses a_{k,k} as a sum over ordered trees with weights, and that expression is not in the cited literature. The inductive proof of the tree formula is clean and self-contained. The recursive coefficient formulas and the Vandermonde algorithm (Prop 2.8) work and give a reasonable computational shortcut. The invariants a_{j,k} are interesting, and the paper honestly flags the open question about their combinatorial meaning.

Now the soft spots, in order of severity. First, Definition 3.1 is under-specified: the bullet constraining children to sum to v applies only to nodes labeled v with 2 < v < k, so label 2 nodes are unconstrained. Literal reading of T_3 gives four trees, not two, and Proposition 3.5 gives -1/180 instead of -1/360. The intended decomposition in Remark 3.3(2) clearly uses the rule for all v > 1; the fix is to replace '2 < v < k' with '1 < v < k'. This must be corrected.

Second, the proof of Theorem 2.1 has a substitution error: r(ζ_{2^{n+2}}) is used where the argument needs ζ_{2^{n+3}}. The structural identity x^{2^{n+1}} p_n(x+1/x) = x^{2^{n+2}}+1 is true (it is the Chebyshev doubling identity), but it is asserted without proof or citation. Both are fixable.

Third, Theorem 2.5's proof defers a key summation interchange to the second author's PhD thesis, which is not accessible. That is a serious gap in a proof of a central theorem; the identity should be written out or cited to a published source.

Finally, the paper's positioning is misleading: it presents the coefficient expansion as field-theoretic, but it is a known Chebyshev result. The authors should acknowledge the connection. This is not a fatal flaw -- the tree formula is still new -- but it changes how the paper should be framed.

For a reader: this is for combinatorialists who like Catalan variants and people working with Chebyshev iterations. The tree formula is a nice classroom example. I would not cite it for the coefficient formulas, but I would cite it for the tree encoding if the definition is fixed.

Verdict: send it to review with a request for major revision. The math is mostly right and the new result is real, but the definition bug and the missing identity must be addressed.

Referee Report

3 major / 5 minor

Summary. The paper studies the polynomial sequence defined by p_0(x)=x^2-2 and p_n(x)=p_{n-1}(x)^2-2. It relates these polynomials to minimal polynomials of generators of 2-power cyclotomic extensions, derives a recursive formula for the coefficients (Theorem 2.1), expresses each even coefficient c_{n,2k} as a finite sum of n-independent invariants a_{j,k} times powers of 2^{2jn} (Theorem 2.5), and finally interprets the diagonal invariants a_{k,k} as weighted sums over labeled ordered trees whose cardinality is Catalan (Proposition 3.5). A Vandermonde-based algorithm for computing the invariants is also supplied (Proposition 2.8).

Significance. If the results are correct, the invariant decomposition and the weighted-Catalan formula are concrete and potentially useful, and the paper provides a reproducible algorithm and several worked examples. The combinatorial interpretation of a_{k,k} is the main advertised contribution. However, the manuscript as written contains a defective definition and some proof gaps that affect the stated results; these issues appear fixable, and the underlying claims seem true.

major comments (3)

[Definition 3.1, Remark 3.3, Proposition 3.5] Definition 3.1 is not the intended definition. The bullet 'For each node labeled v with 2 < v < k' leaves nodes labeled 2 unconstrained, so under a literal reading T_3 contains four trees, not the two drawn in Example 3.2. Consequently Remark 3.3(1), which asserts that |T_k| is the (k-1)-th Catalan number, fails for k=3, and the weighted sum in Proposition 3.5 would give -1/180 instead of a_{3,3}=-1/360. The proof of Proposition 3.5 uses the Catalan decomposition of Remark 3.3(2), which requires every node with label v>1 to have two ordered children whose labels sum to v. Please replace '2 < v < k' with '1 < v < k' (or equivalently '2 <= v < k') and adjust the surrounding text and example accordingly.
[Theorem 2.1, Section 2] The proof of Theorem 2.1 contains an incorrect root-of-unity substitution. The text states r(zeta_{2^{n+2}}) = zeta_4 p_n(t^+_{zeta_{2^{n+2}}}) = 0 and then concludes r(x)=min(zeta_{2^{n+3}},Q)=x^{2^{n+2}}+1. But zeta_{2^{n+2}} is a primitive 2^{n+2}-th root of unity and is not a root of x^{2^{n+2}}+1; the argument works if the evaluation is at zeta_{2^{n+3}} instead. Additionally, the identity x^{2^{n+1}} p_n(x+1/x)=x^{2^{n+2}}+1, equivalently p_n(z+z^{-1})=z^{2^{n+1}}+z^{-2^{n+1}}, is used without proof or citation. Please correct the index and supply a proof or reference for this identity.
[Theorem 2.5, Section 2] The proof of Theorem 2.5 is incomplete as written. The step 'by applying the induction hypothesis and the base change of the sums (see [3, pg 201])' is used to derive the expressions for f_{n-1,k} and g_{n-1,k}. This is a key summation identity on which the induction rests, and the cited source is a PhD thesis that the reader may not be able to access. Please prove the identity in the paper or provide a published reference.

minor comments (5)

[Definition 1.1] The heading 'Definition 1.1 (Lemma)' is confusing; please decide whether this is a definition or a lemma and format accordingly.
[Notation and symbols] The interval notation '/llbracket m,n/rrbracket' appears as raw LaTeX in the notation table; please render it as [m,n] consistently throughout the paper.
[Remark 2.2(5)] Remark 2.2(5) is terse; it would be clearer if the constant term c_{n,0}=2 were stated explicitly before the displayed sum.
[Example 2.9(1)] In Example 2.9(1), the rows of the Vandermonde system correspond to n=2,3,4, but the indexing of the right-hand vector is not explained; please define the n-values explicitly.
[References] Reference [3] is a PhD thesis; if it is freely accessible online, please include a stable URL, otherwise consider replacing it with a published source for the summation identity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the coefficient recursion, invariant expansion, and weighted-Catalan representation are derived from the polynomial recursion and explicit definitions, not from the claimed conclusions.

full rationale

The paper's central derivation is self-contained. Lemma 2.4 follows algebraically from p_n(x)=p_{n-1}(x)^2-2, and Theorem 2.5 introduces the invariants a_{j,k} by explicit recursive definitions and proves c_{n,2k}=sum_j a_{j,k}2^{2jn} by induction; the a_{j,k} are not fitted to the c_{n,2k}. The proof of Theorem 2.1 uses the identity x^{2^{n+1}}p_n(x+1/x)=x^{2^{n+2}}+1, which is true by the Chebyshev doubling property and is an independent mathematical input, not a disguised form of the target theorem. The cyclotomic minimal-polynomial facts cited to the authors' prior work [2] are classical and externally checkable, so they do not constitute load-bearing self-citation circularity. Proposition 3.5 proves a_{k,k}=sum_T prod_v b_v^{delta_{v,T}} by induction using the standard Catalan tree decomposition in Remark 3.3, with weights b_v already fixed in Theorem 2.5. No claimed output reduces to its own input by construction. A separate, non-circularity defect exists: Definition 3.1 does not constrain labeled-2 nodes, so read literally T_3 would contain extra trees and Proposition 3.5 would fail as stated; this is a definitional-correctness issue, not a circularity issue.

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

The central claim rests on the Chebyshev identity (used implicitly), the standard cyclotomic minimal polynomial Phi_{2^{n+3}}(x) = x^{2^{n+2}} + 1, and prior results of the same authors [1, 2]. No free parameters are fitted to data. The tree objects T_k are new combinatorial constructs but not physical entities; they carry no independent-evidence requirement.

assumptions (4)

standard math p_n(z + z^{-1}) = z^{2^{n+1}} + z^{-2^{n+1}} for all n
Implicitly used in the proof of Theorem 2.1 to obtain x^{2^{n+1}} p_n(x + 1/x) = x^{2^{n+2}} + 1; it is the defining Chebyshev doubling identity but is not stated or proved.
standard math min(zeta_{2^{n+3}}, Q) = x^{2^{n+2}} + 1
Used in Theorem 2.1's proof via [2, Corollary 4.2]; this is the standard cyclotomic polynomial Phi_{2^{n+3}}(x).
domain assumption Prior results from the authors' own work [2, Theorem 3.2, Corollary 4.2] on quadratic cyclotomic extensions
Used in Proposition 1.5 and Remark 1.6 for the minimal-polynomial characterization; not proved in this paper.
domain assumption Sum-manipulation identity cited to [3, pg 201] (the second author's PhD thesis)
Used in the proof of Theorem 2.5 to change bases of double sums; the thesis is not publicly available, so the step cannot be independently checked from the paper.

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Cite this review

Pith. "Pith review of A study of a recursive sequence of polynomials revealing weighted Catalan Numbers." pith.science (2026). https://pith.science/paper/XDLCPBJB

@misc{pith2026250113693,
  author       = {Pith},
  title        = {Pith review of: A study of a recursive sequence of polynomials revealing weighted Catalan Numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XDLCPBJB}},
  note         = {Machine review of arXiv:2501.13693}
}

abstract

This paper examines the recursive sequence of polynomials $p_n(x)$, defined by $p_0(x) = x^2 - 2$ and $p_n(x) = p_{n-1}(x)^2 - 2$ for $n \geq 1$. It describes the field-theoretic motivations behind this sequence, derives a recursive formula for its coefficients, and identifies invariants that uncover combinatorial connections, including links to weighted Catalan numbers.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

A note on quadratic cyclotomic extensions

S. Marques and E. Mrema. A note on quadratic cyclotomic ex tensions. arXiv:2210.03563, 2024

work page Pith review arXiv 2024
[2]

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S. Marques and E. Mrema. When is a 2-Power Cyclotomic Exte nsion cyclic? arXiv:2308.08865v2, 2024

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[3]

E. Mrema. Study of cyclotomic extension of degree power o f 2 and classiﬁcation of radical extensions up to isomorphism. PhD thesis, Stellenbosch University, 20 24

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˙A. Björck and V. Pereyra. Solution of Vandermonde systems of equations. Mathematics of computation, 24(112):893-903, 1970

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E. Aziz. Orthogonal polynomials: Jacobi, Hermite and La guerre polynomials. Preprint, 2020

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Benjamin and D

A. Benjamin and D. Walton. Counting on Chebyshev polynom ials. Mathematics Magazine, 82(2):117- 126, 2009

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Journal of Physics A: Mathematical and General , 37(3):657, 2004

On the construction of recurrence relations for the expa nsion and connection coeﬃcients in series of Jacobi polynomials. Journal of Physics A: Mathematical and General , 37(3):657, 2004

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Integral Transforms and Special Functions , 15(1):13-29, 2004

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[9]

A. Hone, L. Jeﬀery and R. Selcoe. On a family of sequences r elated to Chebyshev polynomials. Journal of Integer Sequences 21(7) , 2018

2018

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

A note on quadratic cyclotomic extensions

S. Marques and E. Mrema. A note on quadratic cyclotomic ex tensions. arXiv:2210.03563, 2024

work page Pith review arXiv 2024

[2] [2]

When is a 2-Power Cyclotomic Extension cyclic?

S. Marques and E. Mrema. When is a 2-Power Cyclotomic Exte nsion cyclic? arXiv:2308.08865v2, 2024

work page Pith review arXiv 2024

[3] [3]

E. Mrema. Study of cyclotomic extension of degree power o f 2 and classiﬁcation of radical extensions up to isomorphism. PhD thesis, Stellenbosch University, 20 24

work page

[4] [4]

Björck and V

˙A. Björck and V. Pereyra. Solution of Vandermonde systems of equations. Mathematics of computation, 24(112):893-903, 1970

work page 1970

[5] [5]

E. Aziz. Orthogonal polynomials: Jacobi, Hermite and La guerre polynomials. Preprint, 2020

work page 2020

[6] [6]

Benjamin and D

A. Benjamin and D. Walton. Counting on Chebyshev polynom ials. Mathematics Magazine, 82(2):117- 126, 2009

work page 2009

[7] [7]

Journal of Physics A: Mathematical and General , 37(3):657, 2004

On the construction of recurrence relations for the expa nsion and connection coeﬃcients in series of Jacobi polynomials. Journal of Physics A: Mathematical and General , 37(3):657, 2004

work page 2004

[8] [8]

Integral Transforms and Special Functions , 15(1):13-29, 2004

In the connection coeﬃcients and recurrence relations a rising from expansions in series of Hermite polynomials. Integral Transforms and Special Functions , 15(1):13-29, 2004

work page 2004