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Euler-type recurrences for $t$-color and $t$-regular partition functions

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The 3-colored partition function obeys an infinite family of exact triangular-number recurrences, with corrections built from divisor sums and Hecke traces.

desk verdict A competent extension of the Gomez–Ono–Saad–Singh framework to t-colored and t-regular partitions, whose genuinely new infinite family of p3 recurrences is conditional on two unproved lemmas; the elementary recurrences are solid and correctly derived. read the letter →

arxiv 2412.14344 v2 pith:3B4PP34F submitted 2024-12-18 math.NT math.CO

Tapas Bhowmik , Wei-Lun Tsai , Dongxi Ye This is my paper · ORCID
classification math.NTmath.CO MSC 05A1711F1111F2511P81
keywords Euler-typerecurrence3-coloredpartitiontriangularnumberRankin–CohenbracketHecketracet-regularmodularformfunction
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 establishes Euler-type recurrences for t-colored partitions (t = 2 and 3) and for t-regular partitions of every order. Its central claim is an infinite family of 'triangular number' recurrences for the 3-colored partition function p3(n): for each weight v in {6} ∪ Z_{≥8}, the value p3(n) is expressed in terms of earlier values p3(n − T_k) at triangular numbers T_k, corrected by a divisor-sum term and by the Hecke trace Tr_{2v}(n). The recurrences are proved by forming the v-th Rankin–Cohen bracket of 1/η(z)^3 with η(z)^3, showing it is a holomorphic modular form of weight 2v and level 1, and then comparing Fourier coefficients. Explicit small-weight versions of the recurrences are worked out, with correction terms reducing to divisor sums and to Ramanujan-type coefficients, together with a pentagonal-number recurrence for 2-colored partitions and for all t-regular partitions.

What carries the argument

The central object is the $v$-th Rankin–Cohen bracket $R_v(z):=[1/\eta(z)^3,\eta(z)^3]_v$, a bilinear combination of the weight $-3/2$ form $1/\eta^3$ and the weight $3/2$ form $\eta^3$ and their derivatives. Lemma 2.2 shows $R_v$ is a holomorphic modular form of weight $2v$ and level $1$, with $q$-expansion $$R_v(z)=\sum_{n\ge 0}\left(\sum_{k\ge 0}(-1)^k E_v(n,k)\,p_3(n-T_k)\right)q^n.$$ Since the space of weight-$2v$ level-$1$ modular forms is spanned by the Eisenstein series $E_{2v}$ and the cusp forms $S_{2v}(1)$, comparing coefficients of $q^n$ produces the recurrences once the cuspidal part is identified. The load-bearing identification is Proposition 2.5, which evaluates the Petersson inner product $\langle R_v,f\rangle$ as $\|f\|\cdot D_f$, where $D_f$ is a weighted infinite sum of special values of twisted Dirichlet series associated to $f$; the proof passes through a Maass–Poincaré series representation of $1/\eta^3$, an integral evaluation using M-Whittaker functions, and Euler and Pfaaff–Saalschütz hypergeometric reductions.

What would settle it

Compute the right-hand side of Theorem 1.4 for $v=8$ and $n=5$, using the stated definition of $\operatorname{Tr}_{16}(5)$ with truncated sums over $m$ and over eigenforms, and compare it with $p_3(5)=108$; any nonzero discrepancy would falsify the general recurrence. As a lighter check, recompute $D_\Delta$ for the weight-12 case with the paper's own summation ranges and compare with $\beta_6=-51051/22112$, which tests the convergence assumptions of Proposition 2.5 without invoking higher-weight cusp data.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the generating function identity $$1/(q;q)_\$infty^{3}$ = $q^{{1/8}}$/\eta(z)^3$$ combines with the Jacobi triple product expansion $\eta(z)^3 = q^{1/8}\sum_{k\ge 0}(-1)^k(2k+1)q^{T_k}$ to produce, for every $v\in\{6\}\cup\mathbb{Z}_{\ge 8}$ and every positive integer $n$, the exact recurrence of Theorem 1.4: $$p_3(n)=\frac{1}{E_v(n,0)}\left(-\frac{4v\,E_v(0,0)}{B_{2v}}\,\sigma_{2v-1}(n)+\operatorname{Tr}_{2v}(n)+\sum_{k\ge 1}(-1)^{k+1}E_v(n,k)\,p_3(n-T_k)\right).$$ Here $E_v(n,k)$ are the explicit rational coefficients coming from the Rankin–Cohen bracket expansion, $B_{2v}$ is a Bernoulli number, $\sigma_{m}$ is a divisor sum, and $\operatorname{Tr}_{2v}(n)$ is a weighted sum of Fourier coefficients of normalized Hecke eigenforms of weight $2v$. The paper also proves the explicit small-weight recurrences of Theorem 1.2, the 2-colored recurrence of Theorem 1.1, and the $t$-regular recurrences of Theorem 1.6. The core claim is that comparing the modular-form expansion of the bracket against Eisenstein series and cusp forms yields these recurrences uniformly in $v$, with the cuspidal part exactly a Hecke trace.

Load-bearing premise

Everything beyond the small-weight recurrences rests on Proposition 2.5, which identifies the Petersson inner product with a weighted Dirichlet-series sum; that identification assumes the Poincaré-series representation of $1/\eta^3$ and the hypergeometric reduction are correct (both are quoted rather than proved here), and assumes the double sums defining $D_f$ converge.

Editorial extensions

If this is right

  • For $v\in\{2,3,4,5,7\}$, Theorem 1.2 gives explicit recurrences for $p_3(n)$ whose only arithmetic input beyond earlier partition values is the divisor sum $\sigma_{2v-1}(n)$.
  • For $v\in\{6,8,9,10,11,13\}$, the recurrences gain a correction term $\beta_v\,\tau_{2v}(n)$ from the unique normalized cusp form of weight $2v$.
  • Theorem 1.4 extends the family to every $v\ge 8$, with the cuspidal correction given by the Hecke trace $\operatorname{Tr}_{2v}(n)$, so the recurrences exist for all even weights at least 16 (and weight 12).
  • Theorem 1.6 gives every $t$-regular partition function a pentagonal-number recurrence, with a correction term only when $n=t\,w_j$ for some pentagonal number $w_j$.
  • Lemma 2.1 guarantees $E_v(n,0)\neq 0$ for all $n\ge 1$, so each recurrence genuinely solves for $p_3(n)$.

Reading between the lines

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

  • If the general recurrence holds, the Hecke trace $\operatorname{Tr}_{2v}(n)$ is an effectively computable arithmetic quantity; the paper's $v=6$ numerical check ($D_\Delta(100,700)=-2.308746\ldots$) suggests the defining double sums converge quickly, so truncation bounds for $D_f$ would turn the recurrences into practical algorithms for large $n$.
  • The modular interpretation suggests that $p_3(n)$ may inherit congruence properties from the Hecke eigenvalues, parallel to Ramanujan-type congruences for $p(n)$; reducing the $v=2$ recurrence modulo small primes is a direct way to test this.
  • The Rankin–Cohen bracket construction may extend to other eta quotients $1/\eta^t$ whose reciprocal is a harmonic Maass form of negative weight, yielding analogous recurrences for other colored partition functions beyond $t=2,3$.
  • The main unproved-in-detail steps (the Poincaré-series representation of $1/\eta^3$ and the hypergeometric reduction in Lemma 2.8) are exactly the places where an independent derivation or an explicit error bound would be needed to make the infinite family fully self-contained.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper derives Euler-type recurrences for 2-colored partitions (Theorem 1.1), for all t-regular partitions (Theorem 1.6), and for 3-colored partitions (Theorems 1.2 and 1.4). The method is to apply Rankin--Cohen brackets to 1/η^3 and η^3; comparing Fourier coefficients converts the identity into recurrences involving triangular numbers, divisor functions, and Hecke traces. Theorem 1.2 treats the small weights where the cusp space is zero- or one-dimensional, while Theorem 1.4 states an infinite family valid for weights 2v with v in Z_{≥8} ∪ {6}, with the cuspidal correction expressed as a weighted sum of twisted Dirichlet series over normalized Hecke eigenforms. Theorems 1.1 and 1.6 are proved by elementary q-series coefficient comparison, and the proof of Theorem 1.2 is also direct. The proof of Theorem 1.4 passes through a Maass–Poincaré series identification, a Petersson inner product computation, and a hypergeometric reduction, and the paper currently leaves important parts of that chain unproved or misstated.

Significance. If the missing details are supplied, the paper gives a genuinely new family of exact triangular-number recurrences for p_3(n), parallel to the pentagonal-number framework of Gomez–Ono–Saad–Singh, and it cleanly handles the t-regular case. The elementary recurrences in Theorems 1.1 and 1.6 are immediately usable and their proofs are transparent. The coefficient-comparison strategy in Theorem 1.2 is sound and the stated constants appear internally consistent. However, the full infinite-family claim in Theorem 1.4 is not yet established because it depends on an incorrectly stated multiplier in the Maass–Poincaré identification, an omitted hypergeometric proof, and unaddressed convergence questions. The significance is therefore conditional on those repairs.

major comments (3)
  1. [§2, Lemma 2.4 and Proposition 2.5] The multiplier in the Poincaré series identification appears to be wrong. Under the convention stated in (2.1), η(γz) = ε(γ)(cz+d)^{1/2}η(z), so η^3 has multiplier ε^3 and 1/η^3 has multiplier ε^{-3}. Lemma 2.4 identifies 1/η^3 with P_{[i∞]}(z,8,-3/2,ε^3), and Proposition 2.5 then uses ε^3 for both the δ-factor and η^3. If both factors carry ε^3, the Rankin–Cohen bracket has multiplier ε^6, contradicting Lemma 2.2 and the identity 1/η^3 · η^3 = 1. Concretely, under T: z ↦ z+1, η(z+1)^3 = e^{π i/4}η(z)^3, so 1/η(z+1)^3 = e^{-π i/4} 1/η(z)^3. The Poincaré series and all slash actions in Proposition 2.5 should use ε^{-3} for the 1/η^3 factor. This is load-bearing for the trace formula in Theorem 1.4.
  2. [§2, Lemma 2.8] The proof of Lemma 2.8 is omitted (“we leave all the details here”). The second identity in this lemma is exactly the step that converts the hypergeometric expression for ω_v(n) into the sums E_v(j,m) defining D_f, so Proposition 2.5 and hence the computation of Tr_{2v}(n) in Theorem 1.4 depend on it. A complete derivation, or a precise reference with the exact statement used, must be supplied; saying that it is “easily adopted” from [7] is not sufficient for a referee to verify the central claim.
  3. [§2, Proposition 2.5 and §1, equation (1.9)] Convergence of the relevant infinite objects is asserted rather than proved. The identity (2.2) represents 1/η^3 as an infinite Poincaré series, and the passage from (2.4) to (2.9) interchanges an infinite sum over γ with an integral over a non-compact domain and then interchanges sums over n and m. In addition, D_f in (1.9) is a double sum over j and m of special values D(f; 2v+2j+2m+2), whose convergence is not justified. Since Theorem 1.4 uses D_f as the cuspidal correction, this missing analysis is load-bearing.
minor comments (4)
  1. [§1, Example after Theorem 1.4] The v=6 numerical computation is described as a numerical justification of Theorem 1.4, but it is only a consistency check: S_12 is one-dimensional, so β_6 is already determined by the coefficient comparison in Theorem 1.2, and D_Δ(100,700) agreeing with β_6 tests the truncated series, not the trace formula independently.
  2. [§2, Lemma 2.1] The proof of Lemma 2.1 says that “by induction argument” a prime ℓ ≠ 3 exists dividing 8n−1; the induction is not shown. For example, one can use that 8n−1 ≡ 7 (mod 8), so the number cannot be a power of 3. Adding this sentence would make the argument complete.
  3. [§2, Proposition 2.5] There are several typos in this section: “Peterson” should be “Petersson”, “harmonoic” should be “harmonic”, and “the followings are true” should be “the following are true”.
  4. [References] Reference [7] is an arXiv preprint (arXiv:2411.16968). If the paper relies on Proposition 3.4 and Lemmas 3.12/3.13 of [7] as heavily as it does, the authors should state explicitly which results are quoted from the preprint and verify that the preprint version is stable and publicly available.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the recurrences follow from the identity 1/eta^3·eta^3 = 1, standard q-series, and modular coefficient comparison; deferred lemmas are correctness gaps, not circular inputs.

full rationale

The derivation chain is self-contained in the relevant sense. The recurrences in Theorems 1.1, 1.2, 1.4, and 1.6 are obtained by multiplying the generating-function identity 1 = 1/η^3 · η^3, using the classical q-expansions of (q;q)_∞ and (q;q)_∞^3, and then applying the Rankin–Cohen bracket together with the modularity of R_v established in Lemma 2.2 from standard differential identities and Proposition 2.3. The constants α_v and β_v are not fitted to p3-data: α_v is identified with E_v(0,0) by comparing constant terms, and β_v is determined by comparing the q-coefficient in a one-dimensional cusp space (Section 4), giving the displayed formula β_v = 4vE_v(0,0)/B_{2v} + 3E_v(1,0) − E_v(1,1). The Petersson inner product computation in Proposition 2.5 expresses ⟨[η^{-3},η^3]_v, f⟩ as ||f||D_f, where D_f is explicitly defined as a weighted sum of twisted Dirichlet values in (1.8)–(1.9); thus the Hecke trace Tr_{2v}(n) is not a fitted parameter renamed as a prediction. The numerical check D_Δ(100,700) = −2.308746... is an independent evaluation of the infinite-sum formula against the coefficient-comparison value β6, not an input to the theorem. The paper does contain expository gaps: Lemma 2.4 defers its proof to [7, Prop. 3.4], Lemma 2.8 says 'we leave all the details here', and convergence of the double sums in (1.9) is asserted rather than proved. These are correctness and rigor concerns, but they are not circularity: they import external identities or omit details, and no step of the argument is equivalent by definition to the claimed recurrence, nor is any self-citation load-bearing.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard q-series identities (Jacobi triple product, Euler pentagonal theorem), Rankin-Cohen bracket theory, the structure of level 1 modular forms, and the representation of 1/eta^3 as a Maass-Poincare series taken from the Gomez-Ono-Saad-Singh framework. No free parameters are fit: alpha_v, beta_v, and E_v(n,k) are computed, not tuned. No new entities are introduced; the Maass-Poincare series, twisted Dirichlet series D(f; s), and Hecke traces are standard modular-form constructions.

assumptions (6)
  • standard math Jacobi triple product gives eta^3(q) = sum_{k>=0} (-1)^k(2k+1) q^{T_k}; Euler pentagonal theorem gives (q;q)_infinity = sum_{j in Z} (-1)^j q^{w_j}.
    Used directly for coefficient comparison in (1.4), Theorem 1.1, and Theorem 1.6 (Sections 1 and 3).
  • standard math Rankin-Cohen bracket modularity: [f,g]_v is modular of weight k+l+2v with multiplier product (Proposition 2.3, citing [3]).
    Load-bearing for Lemma 2.2, which converts the bracket of 1/eta^3 and eta^3 into a level-1 weight-2v modular form with trivial multiplier.
  • standard math Structure of level 1 modular forms: M_{2v}(1) = C E_{2v} direct sum S_{2v}(1), with stated dimensions (zero cusp space for v in {2,3,4,5,7}, one-dimensional for v in {6,8,9,10,11,13}).
    Justifies the ansatz R_v = alpha_v E_{2v} + beta_v Delta_{2v} and the eigenform decomposition in Theorem 1.4 (Section 4).
  • domain assumption Representation of 1/eta^3 as the Maass-Poincare series P_{i infinity}(z, 8, -3/2, epsilon^3) (Lemma 2.4, proof deferred to [7, Prop. 3.4]).
    This identification is the entry point for the Petersson inner product computation in Proposition 2.5; the proof is not given in this text.
  • standard math Whittaker integral evaluation [10, 13.23.1] in Lemma 2.7 and the Euler/Pfaaff-Saalschuetz hypergeometric identities in Lemma 2.8.
    These reduce the inner product to the double Dirichlet series D_f; Lemma 2.8 is stated with 'we leave all the details here', so it is an unverified-in-text input to Theorem 1.4.
  • domain assumption Convergence and termwise exchange of sums/integrals in the definitions of D_f and Tr_{2v} for Re(s) >= 2v+2.
    Assumed for the formal manipulations in Proposition 2.5 and Theorem 1.4; stated but not proved in the text.

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Pith. "Pith review of Euler-type recurrences for $t$-color and $t$-regular partition functions." pith.science (2026). https://pith.science/paper/3B4PP34F

@misc{pith2026241214344,
  author       = {Pith},
  title        = {Pith review of: Euler-type recurrences for $t$-color and $t$-regular partition functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3B4PP34F}},
  note         = {Machine review of arXiv:2412.14344}
}
abstract

We give Euler-like recursive formulas for the $t$-colored partition function when $t=2$ or $t=3,$ as well as for all $t$-regular partition functions. In particular, we derive an infinite family of ``triangular number" recurrences for the $3$-colored partition function. Our proofs are inspired by the recent work of Gomez, Ono, Saad, and Singh on the ordinary partition function and make extensive use of $q$-series identities for $(q;q)_{\infty}$ and $(q;q)_{\infty}^3.$

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Reference graph

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