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arxiv: 1906.11978 · v1 · pith:RA2LNVJGnew · submitted 2019-06-27 · 🧮 math.NT

Some Observations on Lambert series, vanishing coefficients and dissections of infinite products and series

James Mc Laughlin This is my paper

Pith reviewed 2026-05-25 14:21 UTC · model grok-4.3

classification 🧮 math.NT
keywords Lambert seriesvanishing coefficientsq-product dissectionsRamanujan 1ψ1 identityinfinite q-productsFine functionq-series identities
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The pith

Results on vanishing coefficients and m-dissections of q-products follow automatically from specializations of an identity derived from Ramanujan's 1ψ1 summation.

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

The paper shows that many earlier theorems establishing the vanishing of coefficients in arithmetic progressions within expansions of certain infinite q-products are obtained directly by choosing specific parameter values in a general identity that comes from a special case of Ramanujan's 1ψ1 summation. The identical method recovers a collection of known results on the m-dissections of these products. The same specialization technique also produces two identities presented as extensions of Ramanujan's work and yields explicit curious q-series identities when applied to other Lambert series, with closed forms stated for arbitrary positive integer m. The ideas are further applied to the Fine function F(a,b;t).

Core claim

Various results on the vanishing of coefficients in arithmetic progressions for certain infinite q-products, as well as results on their m-dissections, follow automatically by specializing parameters in an identity derived from a special case of Ramanujan's 1ψ1 identity. Two identities that may be considered as extensions of two identities of Ramanujan are also derived. Applying similar ideas to certain other Lambert series gives rise to some rather curious q-series identities, such as the given multline identity for any positive integer m, and another involving (aq;q)∞ sum. Applications to the Fine function F(a,b;t) are considered.

What carries the argument

The general identity derived from a special case of Ramanujan's 1ψ1 summation, from which coefficient vanishings and m-dissections follow by parameter specialization.

If this is right

  • All cited vanishing-coefficient theorems for q-products become immediate corollaries of the single general identity.
  • The m-dissections of the products are recovered uniformly by the same specialization process.
  • Two new identities extending Ramanujan's work are obtained without separate derivation.
  • Explicit identities for Lambert series hold for every positive integer m.
  • The Fine function F(a,b;t) admits analogous specializations yielding new relations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Other known q-series identities not treated here may likewise reduce to special cases of the same 1ψ1-derived identity.
  • The unification of vanishing-coefficient and dissection results points to a single mechanism underlying both phenomena.
  • The method may generate additional families of curious identities when applied to further Lambert series.
  • Direct verification of the given multline identity for small m would test consistency with known product expansions.

Load-bearing premise

The general identity remains valid after the parameter specializations that produce the stated coefficient vanishings or dissections, without convergence or other restrictions invalidating the results.

What would settle it

A concrete arithmetic progression in the expansion of one of the q-products where the coefficient predicted by the specialized identity differs from the coefficient established by prior direct proofs.

read the original abstract

Andrews and Bressoud, Alladi and Gordon, and others, have proven, in a number of papers, that the coefficients in various arithmetic progressions in the series expansions of certain infinite $q$-products vanish. In the present paper it is shown that these results follow automatically (simply by specializing parameters) in an identity derived from a special case of Ramanujan's $_1\psi_1$ identity. Likewise, a number of authors have proven results about the $m$-dissections of certain infinite $q$-products using various methods. It is shown that many of these $m$-dissections also follow automatically (again simply by specializing parameters) from this same identity alluded to above. Two identities that mat be considered as extensions of two Identities of Ramanujan are also derived. It is also shown how applying similar ideas to certain other Lambert series gives rise to some rather curious $q$-series identities, such as, for any positive integer $m$, \begin{multline*} {\displaystyle \frac{\left(q,q,a,\frac{q}{a},\frac{b q}{d}, \frac{dq}{b}, \frac{aq}{b d}, \frac{b d q}{a};q\right)_{\infty }} {\left(b,\frac{q}{b},d,\frac{q}{d},\frac{a}{b},\frac{bq}{a},\frac{a}{d},\frac{dq}{a};q\right)_{\infty }}} = \sum _{r=0}^{m-1} q^r \frac{ \left(q^m,q^m,a q^{2 r},\frac{q^{m-2 r}}{a},\frac{b q^m}{d},\frac{d q^m}{b}, \frac{a q^m}{b d},\frac{b dq^m}{a};q^m\right){}_{\infty }} {\left(b q^r,\frac{q^{m-r}}{b},d q^r,\frac{q^{m-r}}{d},\frac{a q^r}{b},\frac{b q^{m-r}}{a},\frac{a q^r}{d}, \frac{dq^{m-r}}{a};q^m\right){}_{\infty }} \end{multline*} and \begin{equation*} (aq;q)_{\infty}\sum_{n=1}^{\infty} \frac{n a^n q^{n}}{(q;q)_n} = \sum_{r=1}^{m}(aq^{r};q^m)_{\infty}\sum_{n=1}^{\infty} \frac{na^n q^{n r}}{(q^m;q^m)_n}. \end{equation*} Applications to the Fine function $F(a,b;t)$ are also considered.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that known results on vanishing coefficients in arithmetic progressions for certain infinite q-products, as well as m-dissections of such products, follow automatically by specializing parameters in a bilateral identity derived from a special case of Ramanujan's 1ψ1 summation formula. It also derives two identities extending Ramanujan's work, obtains curious q-series identities from other Lambert series (including an explicit m-parameter identity and a second summed identity), and considers applications to Fine's function F(a,b;t).

Significance. If the specializations remain valid, the unified derivation from one parent identity offers a systematic and economical route to multiple prior results on coefficient vanishing and dissections, reducing the need for case-by-case proofs. The explicit m-dissection identity and the Lambert-series extensions are concrete contributions that could streamline further work in q-series.

major comments (2)
  1. [Abstract and the derivation of the general identity (prior to the displayed m-dissection formula)] The central claim that vanishing-coefficient and m-dissection statements 'follow automatically' by specialization rests on the parent bilateral series remaining inside its domain of convergence. The manuscript does not verify that the parameter choices (e.g., the substitutions yielding the displayed m-dissection identity or the coefficient-vanishing cases) keep the series inside the annulus |q| < |z| < 1 (with the usual restrictions on a,b,d) for |q|<1; if any specialization exits this region, the coefficient extraction and dissection identities are no longer justified by the 1ψ1 formula.
  2. [The paragraph containing the second displayed equation after the multline identity] The second displayed identity (the summed form involving (aq;q)∞ and the inner sums over n) is presented as following from similar ideas, yet no explicit check is given that the interchange of summation or the extraction of the m-fold dissection is justified under the same convergence constraints that govern the parent 1ψ1 identity.
minor comments (2)
  1. [Abstract] Abstract: 'mat be' should read 'may be'.
  2. [The multline identity] The notation for the infinite products in the displayed identities is dense; adding a brief reminder of the standard (a;q)∞ definition at first use would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for highlighting the need to verify convergence domains explicitly. We will revise the manuscript to address both major comments by adding the required checks for the parameter specializations and summation interchanges. This strengthens the justification without altering the main results.

read point-by-point responses

  1. Referee: [Abstract and the derivation of the general identity (prior to the displayed m-dissection formula)] The central claim that vanishing-coefficient and m-dissection statements 'follow automatically' by specialization rests on the parent bilateral series remaining inside its domain of convergence. The manuscript does not verify that the parameter choices (e.g., the substitutions yielding the displayed m-dissection identity or the coefficient-vanishing cases) keep the series inside the annulus |q| < |z| < 1 (with the usual restrictions on a,b,d) for |q|<1; if any specialization exits this region, the coefficient extraction and dissection identities are no longer justified by the 1ψ1 formula.

    Authors: We agree that explicit verification is necessary. In the revised manuscript, immediately following the general bilateral identity, we will insert a paragraph confirming that all specializations used (e.g., for vanishing coefficients where z = q^k for integer k with |q|<1 implying |q| < |z| <1 when appropriate, and for m-dissections with z a suitable root of unity times q-power, along with the listed a,b,d) remain strictly inside |q| < |z| < 1. These choices satisfy the annulus condition under |q|<1, justifying the applications of the 1ψ1 formula. revision: yes

  2. Referee: [The paragraph containing the second displayed equation after the multline identity] The second displayed identity (the summed form involving (aq;q)∞ and the inner sums over n) is presented as following from similar ideas, yet no explicit check is given that the interchange of summation or the extraction of the m-fold dissection is justified under the same convergence constraints that govern the parent 1ψ1 identity.

    Authors: We concur that an explicit justification for the interchange and dissection extraction is warranted. In revision, we will add a short paragraph after the second identity explaining that the double sums converge absolutely in the region |q|<1 with the chosen parameters (ensuring |aq^r| <1 etc.), permitting term-by-term rearrangement and m-fold extraction via the same annulus condition as the parent identity. If needed, we can reference standard q-series absolute convergence arguments. revision: yes

Circularity Check

0 steps flagged

Derivations follow from external Ramanujan 1ψ1 identity via parameter specialization; no circular reduction

full rationale

The paper starts from a special case of Ramanujan's prior 1ψ1 summation (an independent, externally established result) to obtain a general bilateral identity, then extracts vanishing-coefficient and m-dissection statements by direct substitution of parameters. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or load-bearing self-citation chain; the source identity is not authored by the present writer and is not itself derived within the paper. The listed extensions of Ramanujan identities and Lambert-series applications are likewise obtained by the same specialization process. Convergence-domain questions affect validity but do not create circularity in the derivation structure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of Ramanujan's 1ψ1 summation as a background result and the assumption that parameter specializations preserve the identity without additional analytic conditions.

axioms (1)
  • standard math Ramanujan's 1ψ1 summation identity holds
    The paper states that the general identity is derived from a special case of it.

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Reference graph

Works this paper leans on

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