Generating Functions and Congruences for Some Partition Functions Related to Mock Theta Functions

Nayandeep Deka Baruah; Nilufar Mana Begum

REVIEW 3 major objections 4 minor 31 references

Generating Functions and Congruences for Some Partition Functions Related to Mock Theta Functions

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves new exact q-series generating functions for partition functions tied to the third-order mock theta functions $\omega(q)$ and $\nu(q)$, and derives new divisibility families modulo powers of 5.

desk verdict A solid, within-subfield paper with two fully proved generating-function theorems and a real soft spot: the unproved identity (1.36) that carries the new high-power congruences. read the letter →

arxiv 1908.07741 v1 pith:44NJ62RT submitted 2019-08-21 math.NT

Nayandeep Deka Baruah , Nilufar Mana Begum This is my paper · ORCID

classification math.NT MSC 11P8305A1505A17

keywords partitioncongruencesmallestpartsfunctionmockthetagenerating5-dissectionRogers-Ramanujancontinuedfractioneta-quotienttotallysymmetricplanepartitions

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 establishes new exact generating functions for four partition functions attached to the third-order mock $\theta$ functions $\omega(q)$ and $\nu(q)$: $p_\nu(50n+8)$, $p_\omega(40n+12)$, $\mathrm{spt}_\omega(10n+3)$, $\mathrm{spt}_\omega(50n+23)$, $\mathrm{spt}_\omega(10n+5)$, and $\mathrm{spt}_\omega(50n+25)$. Each is written as a finite sum of explicit eta-quotients with integer coefficients. From these formulas the authors derive new congruences modulo powers of 5, the strongest being $p_\nu(6250n+5208)\equiv 0 \pmod{125}$ and the family $\mathrm{spt}_\omega(5^{2k+\ell-1}(10n+5))\equiv \mathrm{spt}_\omega(5^{\ell-1}(10n+5))\pmod{5^\ell}$ for $\ell=1,\dots,6$. A sympathetic reader would care because these are exact finite descriptions of counting functions whose divisibility behavior had previously been known only through scattered congruences, and the method reduces each congruence to an algebraic extraction using a fixed set of 5-dissection identities.

What carries the argument

The working tool is a 5-dissection calculus built on the Rogers-Ramanujan continued fraction: writing $T(q)=q^{1/5}/R(q)$, the identities in Lemma 2.1 express $E_1$, $1/E_1$, $\phi(-q)$, and a related eta-quotient in powers of $T(q)$, letting one extract the coefficient of $q^{5n+r}$ in a generating function. Two further identities from the authors' earlier work, Lemmas 2.2 and 2.3, link $T(q)$, $T(q^2)$, and the eta quotient $K=E_2 E_5^5/(E_1 E_{10}^5)$; these are applied repeatedly to rearrange and reduce the extracted series into the compact eta-quotient forms of the theorems.

What would settle it

For $n=0$, compute $p_\nu(8)$ directly from the definition (partitions of 8 with distinct parts and each odd part less than twice the smallest part) and compare it with the constant coefficient of the right-hand side of Theorem 1.1; then repeat for $n=1,2$ against the coefficients of $q$ and $q^2$, which would test $p_\nu(58)$ and $p_\nu(108)$. A mismatch at any of these first three values would refute the identity, and the congruence $p_\nu(6250n+5208)\equiv 0 \pmod{125}$ can be tested independently by enumerating the relevant partitions for $n=0,1,2$.

Watch

Extended reading notes

Core claim

On the paper's own terms: the sequence $p_\nu(50n+8)$ can be written exactly as $5\bigl(E_2^5 E_4^5/(E_1^6 E_{10}^2) + 160 q E_2^{11} E_4^5/E_1^{14} + 2000 q^2 E_2^{11} E_4^{10}/E_1^{20}\bigr)$ (Theorem 1.1), the sequence $p_\omega(40n+12)$ has the nine-term eta-quotient expansion of Theorem 1.4, and the two smallest-parts functions satisfy the explicit generating functions of Theorems 1.7 and 1.8. These are not asymptotic or congruence-only statements: they are exact $q$-series equalities, and each corollary divisibility claim follows by reducing the right-hand side modulo a power of 5 and extracting a specific residue class. The same identities yield the previously known congruences (1.31) and (1.32), the new Corollary 1.9, and Theorem 1.3, which translates the $p_\nu$ result into a congruence for 1-shell totally symmetric plane partitions. The proofs are self-contained in the sense that every new step is an application of 5-dissections and eta-quotient manipulations, starting from known generating functions for the base residue classes.

Load-bearing premise

The quoted identities in Lemmas 2.2 and 2.3 are taken from the authors' earlier paper without proof in this one, and if either is incorrect, the new generating functions and the congruences built on them do not follow.

Editorial extensions

If this is right

If Theorem 1.1 is correct, the congruence ladder for $p_\nu(50n+8)$ extends uniformly: reducing (1.13) modulo 5 and 25 recovers (1.12), and the same formula yields $p_\nu(6250n+5208)\equiv 0 \pmod{125}$.
Theorem 1.7 gives back the known congruences (1.31) and (1.32), previously proved with modular forms, by an elementary generating-function route.
Theorem 1.8 and Corollary 1.9 produce infinite families of congruences for $\mathrm{spt}_\omega$ along arithmetic progressions, including iterations of $\mathrm{spt}_\omega(250n+125)\equiv \mathrm{spt}_\omega(10n+5)\pmod{5}$.
Since $p_\nu(2n)=f(6n+1)$, Theorem 1.1 yields an exact generating function for $f(150n+25)$ and the new congruence $f(18750n+15625)\equiv 0 \pmod{125}$.

Reading between the lines

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

The same dissection-and-extract routine that proves Theorem 1.1 and Theorem 1.4 should, with more labor, produce explicit generating functions for $p_\nu(250n+208)$ and $p_\omega(200n+92)$, giving a ladder of congruences modulo 625 and higher.
The pattern in Corollary 1.9 suggests a general theorem: $\mathrm{spt}_\omega(5^{2k+\ell-1}(10n+5))\equiv \mathrm{spt}_\omega(5^{\ell-1}(10n+5))\pmod{5^\ell}$ for all $\ell$; the paper leaves this as Conjecture 1.10, so a proof would follow if the same reduction pattern persists at every level.
Because $p_\nu(2n)=f(6n+1)$, the new $p_\nu$ formulas translate directly into enumerative information about 1-shell totally symmetric plane partitions, connecting the mock-theta congruences to plane-partition geometry.
A direct computation of the first few coefficients from (1.13) and (1.35), independent of the lemmas, would confirm the extraction chain and could be done by generating all partitions for $n$ up to a few hundred.

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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

The useful content here is real. Theorems 1.1 and 1.4 give new exact generating functions for p_nu(50n+8) and p_omega(40n+12), and the proofs are written out in enough detail that I can follow the 5-dissection machinery and check the algebra. Theorem 1.7's generating function for spt_omega(10n+3) and (1.34) for spt_omega(50n+23) are also plausible and the proof is sketched honestly, though with some 'omitting details' in the extraction step. The paper is careful about credit: it notes that (1.31) and (1.32) were already proved by Wang and Yang, and Corollary 1.5 is attributed to Xia. The reliance on Lemmas 2.2–2.3 from the authors' earlier paper is legitimate; those are published, traceable identities, and self-citation here is not a red flag.

The soft spot is exactly where the stress test points. Theorem 1.8's identity (1.36) is not proved at all—the text says it 'can be proved in a similar way' and omits the proof. That identity is a large eta-quotient with enormous coefficients, and every congruence in Corollary 1.9, including the new mod 5^ℓ families, is derived from it by iterated 5-dissection. If any coefficient or exponent in (1.36) is wrong, those congruence families could collapse. This is not a manufactured flaw; the authors flag the omission themselves. It is also not fatal to the whole paper, because Theorems 1.1 and 1.4 are proved in full and stand independently, and (1.35) in Theorem 1.8 is proved. But as it stands, the paper's headline high-power congruence claims rest on an unverified assertion.

I also want to note that the derivation of the mod 5^4, 5^5, and 5^6 congruences in Corollary 1.9 is largely recorded as a list of successive generating functions with 'can be proved in a similar fashion.' That is the kind of thing that is routine for the authors but not independently checkable by the reader without substantial work. A referee should insist on at least one fully worked iteration, or a verification of (1.36) via a computer algebra system.

Who is this for? Specialists in partition congruences and mock theta functions. They will get value from the explicit generating functions and the congruence families, and they will be able to fill in the omitted algebra. The paper deserves a serious referee and probably acceptance after the missing proof of (1.36) is supplied, not a desk rejection. I would take it for review, but I would send it back for a revision that includes that proof or a computer-checked verification.

Referee Report

3 major / 4 minor

Summary. The paper derives exact q-series generating functions for the partition functions p_nu, p_omega and the smallest-parts functions spt_omega, spt_omega at certain arithmetic progressions, using 5-dissections together with identities from the authors' earlier paper [8]. The main results are Theorem 1.1 for p_nu(50n+8), Theorem 1.4 for p_omega(40n+12), Theorem 1.7 for spt_omega(10n+3) and spt_omega(50n+23), and Theorem 1.8 for spt_omega(10n+5) and spt_omega(50n+25). From these generating functions the authors deduce new congruences modulo powers of 5, including Corollary 1.2, Corollary 1.5, and the families in Corollary 1.9. The proofs are elementary but computation-heavy; Theorems 1.1 and 1.4 are written out in detail, while parts of Theorems 1.7 and 1.8 and Corollary 1.9 are stated with omitted details.

Significance. If the identities and congruences are correct, they are explicit, nontrivial additions to the literature on partition functions associated with mock theta functions and on smallest-parts functions. The paper gives concrete and falsifiable statements, and the detailed proofs of Theorems 1.1 and 1.4 are checkable by the methods of the field. The substantial new congruences in Corollary 1.9 depend essentially on identity (1.36), whose proof is omitted; as written, the manuscript therefore does not fully establish its headline claims. The paper is nonetheless well organized and the computational work is impressive, provided the missing derivations can be supplied.

major comments (3)

[Section 6, Eq. (1.36)] The proof of Theorem 1.8 states that (1.36) 'can be proved in a similar way, therefore we omit the proof.' This is a 17-term eta-quotient identity with coefficients as large as 1024 x 10^18, and it is the sole input for every congruence in Corollary 1.9, including the new families spt_omega(5^{2k+l-1}(10n+5)) congruent to spt_omega(5^{l-1}(10n+5)) modulo 5^l. A single coefficient or exponent error in (1.36) could change the residues obtained by subsequent 5-dissections and invalidate the congruence claims. Please provide a complete proof or a verifiable derivation of (1.36) before acceptance.
[Section 5, Theorem 1.7] The derivation of (1.34) relies on four displayed dissections introduced by the phrase 'omitting details.' In particular, the third and fourth displayed dissections have many large coefficients, and these computations are exactly what produce (1.34) and the subsequent congruences (1.31) and (1.32). The reader cannot check these steps from the manuscript as it stands. Please supply the intermediate dissections or an accompanying reproducible computation.
[Section 6, Corollary 1.9] The cases l = 4, 5, 6 of (1.37) are not proved; the text says 'The remaining cases of (1.37) can be proved in a similar fashion' and then records only successive generating functions. Since (1.39) and (1.40) are new congruence statements, they require either a proof or an explicit statement that they are computer-assisted and verified to a specified order. Please provide the missing details or clearly separate the proved cases from the conjectural ones.

minor comments (4)

[Section 1] The name 'Wladherr' following Eq. (1.5) should be 'Waldherr'.
[Section 2, Eq. (2.5)] The notation (q15,q35,q50;q50)_infty is used without definition; please define the multiple-parameter product notation or replace it with an equivalent product expression.
[Section 6, displayed equations near (6.7)] There are several typographical slips, such as 'spt_omega (1250n + 625))qn' with a double parenthesis, which should be corrected.
[Section 6, proof of Corollary 1.9] The transition 'from (1.36) and (2.3)' in (6.4) is terse; it would help the reader if the congruence modulus being reduced at each stage were stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new generating functions and congruences are derived from external known inputs and auxiliary published identities, not assumed as inputs.

full rationale

The derivation chain starts from externally established generating functions, such as Wang's (1.8), (1.9), (1.22), and (1.23), and Xia's (1.11), and then applies standard 5-dissections (Lemma 2.1, from Berndt) together with Lemmas 2.2 and 2.3 quoted from the authors' prior paper [8]. The lemmas from [8] are auxiliary eta-quotient and Rogers-Ramanujan continued fraction identities; they are parameter-free, published with proofs in [8], and do not contain or presuppose any of the target generating functions or congruences. The paper's new identities are obtained by explicit coefficient extraction and algebraic simplification, not by assuming the desired conclusion. The congruences in Corollaries 1.2, 1.5, and 1.9 are consequences of the derived generating functions after reducing modulo powers of 5; they are not fitted or renamed inputs. The only notable weakness is the omitted proof of identity (1.36) and the omitted details for later cases of (1.37), but an omitted proof is a completeness or correctness risk, not circularity. No equation in the paper is equivalent by construction to a claim it is supposed to derive, and the self-citations are transparent citations to published auxiliary results rather than circular load-bearing support.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivations depend on standard q-series identities and on identities from the authors' prior paper [8]. No free parameters or invented entities are introduced. The main burden is the reliance on Lemmas 2.2 and 2.3 from [8] without proof in this paper.

assumptions (5)

standard math Jacobi triple product identity, used for Ramanujan's theta functions phi(-q) and psi(q) product formulas.
The product representations in (2.1) and (2.2) are standard and cited to Berndt's books.
standard math 5-dissections (2.3), (2.4), (2.5) and identity (2.6) for the Rogers-Ramanujan continued fraction (Lemma 2.1).
These are classical identities recorded from Berndt's books; used throughout Sections 3-6.
domain assumption Lemmas 2.2 and 2.3 from the authors' previous paper [8].
The identities for x=T(q), y=T(q^2) and the eta-type quotients are quoted from [8] without proof and are used in every theorem's derivation.
standard math Identity (3.4) from Ramanujan's Lost Notebook, relating psi^2(q)/psi^2(q^5) to R(q)R^2(q^2).
Used only in the proof of Corollary 1.2; cited to [26] and [2].
standard math Euler's pentagonal number theorem and Jacobi's identity E_1^3 = sum (-1)^k(2k+1)q^{k(k+1)/2} (Eq. (1.25)).
Used to derive old congruences as motivating examples; not load-bearing for new results.

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Cite this review

Pith. "Pith review of Generating Functions and Congruences for Some Partition Functions Related to Mock Theta Functions." pith.science (2026). https://pith.science/paper/44NJ62RT

@misc{pith2026190807741,
  author       = {Pith},
  title        = {Pith review of: Generating Functions and Congruences for Some Partition Functions Related to Mock Theta Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/44NJ62RT}},
  note         = {Machine review of arXiv:1908.07741}
}

abstract

Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third order mock theta functions $\omega(q)$ and $\nu(q)$. In this paper, we find several new exact generating functions for those partition functions as well as the associated smallest parts functions and deduce several new congruences modulo powers of 5.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [8]

N. D. Baruah and N. M. Begum, Exact generating functions for t he number of partitions into distinct parts, Int. J. Number Theory 14 (2018) 1995–2011

work page 2018

[2] [1]

G. E. Andrews, The number of smallest parts in the partitions of n, J. Reine Angew. Math. 624 (2008) 133–142

work page 2008

[3] [2]

G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook, Part I (Springer, New York, 2005)

work page 2005

[4] [3]

G. E. Andrews, S. H. Chan and B. Kim, The odd moments of ranks a nd cranks. J. Combin. Theory, Ser. A. 120 (2013) 77–91

work page 2013

[5] [4]

G. E. Andrews, F. G. Garvan and J. Liang, Combinatorial interpr etations of congruences for the spt-function, Ramanujan J. 29 (2012) 321–338

work page 2012

[6] [5]

G. E. Andrews, A. Dixit and A. J. Yee, Partitions associated with t he Ramanujan/Watson mock theta functions ω (q), ν (q) and φ(q), Res. Number Thoery 1 (2015) Art. 19, 25 pp

work page 2015

[7] [6]

G. E. Andrews, A. Dixit, D. Schultz and A. J. Yee, Overpartitions related to the mock theta function ω (q), Acta Arith. 181 (2017) 253–286

work page 2017

[8] [7]

G. E. Andrews, D. Passary, J. A. Sellers and A. J. Yee, Congrue nces related to the Ramanu- jan/Watson mock theta functions ω (q) and ν(q), Ramanujan J. 43 (2017) 347–357

work page 2017