Transformation Formulae and Applications for Double Lambert Series

Rong chen; Tianjian Xu

arxiv: 2605.28393 · v2 · pith:54JICCXNnew · submitted 2026-05-27 · 🧮 math.NT

Transformation Formulae and Applications for Double Lambert Series

Rong Chen , Tianjian Xu This is my paper

Pith reviewed 2026-06-29 10:10 UTC · model grok-4.3

classification 🧮 math.NT

keywords double Lambert seriestransformation formulaeq-seriesidentitiespartition conjecturesLambert series reductions

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The pith

Double Lambert series reduce to single Lambert series via new transformation formulae.

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

The paper establishes several identities and transformation relations for double Lambert series. These relations allow certain double series to be rewritten as single Lambert series. This provides tools for proving identities connected to recent conjectures in q-series and partitions. A reader would care because these reductions simplify complex expressions that arise in modern number theory research.

Core claim

The authors derive transformation formulae for double Lambert series that reduce them to single Lambert series, apply these to obtain identities related to conjectures of Andrews, Dixit, Schultz, and Yee, and of Amdeberhan, Andrews, and Ballantine, and give a new proof of one result by Amdeberhan, Andrews, and Ballantine.

What carries the argument

Transformation relations that convert double Lambert series into single Lambert series.

If this is right

Certain double Lambert series can be expressed using single Lambert series.
New identities are derived from the conjectures of Andrews, Dixit, Schultz, and Yee.
A new proof is provided for a result of Amdeberhan, Andrews, and Ballantine.
The formulae serve as tools for handling double series in q-series identities.

Where Pith is reading between the lines

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

These transformations could apply to other types of multiple series beyond the ones considered.
Simplifications might extend to generating functions in combinatorial number theory.
Further reductions could lead to closed forms for additional partition-related identities.

Load-bearing premise

The double Lambert series allow the described algebraic manipulations and reductions without extra convergence or domain restrictions.

What would settle it

Evaluating a specific double Lambert series using the transformation and comparing it to direct computation to check for equality.

read the original abstract

In this paper, we study a class of double Lambert series and establish several identities and transformation relations for them. These formulae provide useful tools for reducing certain double Lambert series to single Lambert series. As applications, we derive identities related to recent conjectures of Andrews, Dixit, Schultz, and Yee, and of Amdeberhan, Andrews, and Ballantine. We also propose a new proof of a result of Amdeberhan, Andrews, and Ballantine.

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.

Desk Editor's Note private letter to a colleague

The paper gives transformation formulae that reduce some double Lambert series to single ones and applies them to a few recent partition conjectures, but the abstract alone leaves the actual strength of the claims hard to judge.

read the letter

The main point is that the authors claim several new identities and transformation relations for a class of double Lambert series, with the key feature being reductions that turn certain double series into single ones. They then use these to produce identities tied to conjectures from Andrews, Dixit, Schultz, Yee and from Amdeberhan, Andrews, Ballantine, plus a new proof of one existing result from the latter group.

The work does something concrete by supplying explicit reduction tools in an area where double sums often appear. If those reductions hold cleanly, they could save time for people who need to simplify expressions in q-series calculations. The link to specific conjectures also gives the paper a clear target rather than floating as pure technique.

The soft spots are mostly about scope and verification. The abstract does not display the actual formulae or the steps, so it is not possible to check whether the manipulations require unstated convergence conditions or hold only for narrow parameter ranges. The claim that the double series admit the described algebraic reductions is the load-bearing assumption, and without the derivations it stays untested. The novelty also cannot be confirmed from the summary; it may be a useful extension of standard Lambert series methods rather than a sharp departure. Citation patterns look typical for the subfield, but dependence on prior self-work cannot be assessed here.

This is a paper for specialists already working with Lambert series and partition identities who might need a reduction for a particular problem. A reader outside that niche will not find broad implications. It deserves a serious referee because the applications to named conjectures give it enough substance to justify checking the proofs in detail, even if the overall reach stays inside number theory. I would send it to peer review rather than desk reject on the abstract.

Referee Report

2 major / 0 minor

Summary. The paper studies a class of double Lambert series, establishes several identities and transformation relations allowing reduction of certain double Lambert series to single Lambert series, applies these to derive identities related to conjectures of Andrews-Dixit-Schultz-Yee and Amdeberhan-Andrews-Ballantine, and gives a new proof of a result of the latter authors.

Significance. If the claimed transformations are valid and correctly derived, they would supply practical reduction tools for q-series manipulations in partition theory, potentially aiding resolution or verification of the cited conjectures and adding an alternative proof to the literature.

major comments (2)

Abstract and introduction: no explicit transformation formulae, convergence conditions, or domain restrictions are stated, preventing verification that the algebraic manipulations hold without additional unstated assumptions on the double series.
Applications section: the claimed derivations of identities for the Andrews et al. conjectures and the new proof of the Amdeberhan-Andrews-Ballantine result cannot be assessed for correctness or novelty because no intermediate steps, cited prior results, or explicit reductions are provided in the available text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major point below and will revise the manuscript accordingly to improve clarity and verifiability.

read point-by-point responses

Referee: Abstract and introduction: no explicit transformation formulae, convergence conditions, or domain restrictions are stated, preventing verification that the algebraic manipulations hold without additional unstated assumptions on the double series.

Authors: We agree that the abstract and introduction would benefit from explicitly stating the main transformation formulae, along with the relevant convergence conditions and domain restrictions (such as |q| < 1 and appropriate bounds on the summation indices). This will make the assumptions underlying the algebraic manipulations transparent from the outset. We will revise both sections to include these details. revision: yes
Referee: Applications section: the claimed derivations of identities for the Andrews et al. conjectures and the new proof of the Amdeberhan-Andrews-Ballantine result cannot be assessed for correctness or novelty because no intermediate steps, cited prior results, or explicit reductions are provided in the available text.

Authors: The full manuscript contains the derivations, including intermediate steps, explicit reductions to single Lambert series, and citations to the relevant prior results of Andrews-Dixit-Schultz-Yee and Amdeberhan-Andrews-Ballantine. However, we acknowledge that the presentation may not have been sufficiently detailed for independent verification. In the revision we will expand the applications section to include more explicit step-by-step reductions and clearer cross-references to the cited conjectures and theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract and context describe establishing identities and transformations for double Lambert series to reduce them to single series, with applications to external conjectures. No equations, proofs, self-citations, or derivation steps are provided in the available text that would allow identification of any reduction by construction, fitted inputs renamed as predictions, or load-bearing self-citations. Without explicit load-bearing steps that equate outputs to inputs, the claimed results cannot be shown to be circular; the work is treated as self-contained on the given information.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5589 in / 934 out tokens · 31048 ms · 2026-06-29T10:10:59.065433+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A Generalization of the Amdeberhan-Andrews-Ballantine Conjecture
math.NT 2026-06 unverdicted novelty 6.0

The authors prove an identity generalizing the Amdeberhan-Andrews-Ballantine conjecture on double Lambert series using coefficients from the generalized divisor function σ_k(n).

Reference graph

Works this paper leans on

11 extracted references · 4 canonical work pages · cited by 1 Pith paper · 3 internal anchors

[1]

These formulae provide useful tools for reducing certain double Lambert series to single Lambert series

equation* equation* equation equation [1] http://www.ams.org/mathscinet-getitem?mr= #1 MR #1 Im Ord ord [1] #1 [1] #1 U document [Transformation Formulae and Applications for Double Lambert Series] Transformation Formulae and Applications for Double Lambert Series Rong Chen Department of Mathematics, Shanghai Normal University, People's Republic of China ...

2010
[2]

Amdeberhan, G

T. Amdeberhan, G. E. Andrews and C. M. Ballantine, Lambert series and double Lambert series, J. Combin. Theory Ser. A 221 (2026), Paper No. 106154, 22 pp.; MR5000708

2026
[3]

G. E. Andrews, A. Dixit, D. Schultz, and A. J. Yee, Overpartitions related to the mock theta function (q) , Acta Arith. 181 (2017), no. 3, 253--286; MR3732919

2017
[4]

G. E. Andrews, A. Dixit and A. J. Yee, Partitions associated with the Ramanujan/Watson mock theta functions (q) , (q) and (q) , Res. Number Theory 1 (2015), Paper No. 19, 25 pp.; MR3501003

2015
[5]

B. C. Berndt, Ramanujan's notebooks. Part III , Springer, New York, 1991; MR1117903

1991
[6]

S. P. Cui and D. Tang, Identities and transformations for Lambert series and double Lambert series, preprint, https://arxiv.org/pdf/2604.08839, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[7]

On the Double Lambert Series Conjecture of Andrews-Dixit--Schultz-Yee

Q. Fang, On the double Lambert series conjecture of Andrews-Dixit-Schultz-Yee, preprint, https://arxiv.org/pdf/2604.06242, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[8]

Gasper and M

G. Gasper and M. Rahman, Basic hypergeometric series , second edition, Encyclopedia of Mathematics and its Applications, 96, Cambridge Univ. Press, Cambridge, 2004; MR2128719

2004
[9]

Two Proofs of a Conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series and a new Representation for $E_2(q)$

R. Kumar and A. Singh, On a conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series, preprint, https://doi.org/10.48550/arXiv.2605.21163, 2026

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.21163 2026
[10]

J. H. Lambert, Anlage zur Architectonic, oder Theorie des ersten und des einfachen in der philosophischen und mathematischen Erkenntnis, Vol. 2, Johann Friedrich Hartenoch, Riga, 1771, Philosophische Schriften, vol. 4, Georg Olm, Hildesheim, 1965

1965
[11]

M. D. Schmidt, A catalog of interesting and useful Lambert series identities, preprint, https://doi.org/10.48550/arXiv.2004.02976, 2020

work page doi:10.48550/arxiv.2004.02976 2004

[1] [1]

These formulae provide useful tools for reducing certain double Lambert series to single Lambert series

equation* equation* equation equation [1] http://www.ams.org/mathscinet-getitem?mr= #1 MR #1 Im Ord ord [1] #1 [1] #1 U document [Transformation Formulae and Applications for Double Lambert Series] Transformation Formulae and Applications for Double Lambert Series Rong Chen Department of Mathematics, Shanghai Normal University, People's Republic of China ...

2010

[2] [2]

Amdeberhan, G

T. Amdeberhan, G. E. Andrews and C. M. Ballantine, Lambert series and double Lambert series, J. Combin. Theory Ser. A 221 (2026), Paper No. 106154, 22 pp.; MR5000708

2026

[3] [3]

G. E. Andrews, A. Dixit, D. Schultz, and A. J. Yee, Overpartitions related to the mock theta function (q) , Acta Arith. 181 (2017), no. 3, 253--286; MR3732919

2017

[4] [4]

G. E. Andrews, A. Dixit and A. J. Yee, Partitions associated with the Ramanujan/Watson mock theta functions (q) , (q) and (q) , Res. Number Theory 1 (2015), Paper No. 19, 25 pp.; MR3501003

2015

[5] [5]

B. C. Berndt, Ramanujan's notebooks. Part III , Springer, New York, 1991; MR1117903

1991

[6] [6]

S. P. Cui and D. Tang, Identities and transformations for Lambert series and double Lambert series, preprint, https://arxiv.org/pdf/2604.08839, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[7] [7]

On the Double Lambert Series Conjecture of Andrews-Dixit--Schultz-Yee

Q. Fang, On the double Lambert series conjecture of Andrews-Dixit-Schultz-Yee, preprint, https://arxiv.org/pdf/2604.06242, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[8] [8]

Gasper and M

G. Gasper and M. Rahman, Basic hypergeometric series , second edition, Encyclopedia of Mathematics and its Applications, 96, Cambridge Univ. Press, Cambridge, 2004; MR2128719

2004

[9] [9]

Two Proofs of a Conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series and a new Representation for $E_2(q)$

R. Kumar and A. Singh, On a conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series, preprint, https://doi.org/10.48550/arXiv.2605.21163, 2026

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.21163 2026

[10] [10]

J. H. Lambert, Anlage zur Architectonic, oder Theorie des ersten und des einfachen in der philosophischen und mathematischen Erkenntnis, Vol. 2, Johann Friedrich Hartenoch, Riga, 1771, Philosophische Schriften, vol. 4, Georg Olm, Hildesheim, 1965

1965

[11] [11]

M. D. Schmidt, A catalog of interesting and useful Lambert series identities, preprint, https://doi.org/10.48550/arXiv.2004.02976, 2020

work page doi:10.48550/arxiv.2004.02976 2004