Identities and transformations for Lambert series and double Lambert series

Abstract: We establish two identities for Lambert series and double Lambert series, thereby resolving conjectures of Andrews, Dixit, Schultz and Yee (Acta Arith.~181:253--286, 2017), as well as Amdeberhan, Andrews and Ballantine (J Combin Theory Series A 221:106154, 2026). The proofs are based on classical transformations in the theory of infinite series together with a systematic rearrangement of double Lambert series.

• The paper resolves conjectures on the parity structures of double and single Lambert series through rigorous analytic identities.
• The methodology employs auxiliary series decompositions, index shifts, and q-series product formulas to systematically simplify complex double sums.
• The derived identities provide a framework for analyzing partition congruences and advancing applications in modular and mock modular forms.

Identities and Parity Structures in Lambert Series and Double Lambert Series

Introduction and Context

This paper presents explicit analytic identities involving Lambert series and their generalizations, addressing conjectures of significant interest in combinatorics and $q$-series. Specifically, the authors resolve conjectures formulated by Andrews, Dixit, Schultz, and Yee ("Acta Arith.", 2017) and Amdeberhan, Andrews, and Ballantine ("J. Combin. Theory Ser. A", 2026), both concerning the structural properties and parity phenomena of series expansions that play a pivotal role in partition theory, overpartition identities, and connections to modular and mock modular forms.

Lambert series, defined as $\sum_{n=1}^\infty a_n q^n / (1-q^n)$ for a sequence $\{a_n\}$, appear as generating functions for classical objects such as the divisor function and in expressible modular forms and quasimodular forms. Recent generalizations have led to the study of double Lambert series with two indices and variable dependencies, as well as rational function prescriptions for the coefficients. These expansions elucidate partition-theoretic and modular connections and motivate analytic and combinatorial conjectures.

Main Results

Two central identities are rigorously established:

1. Resolution of the ADSY Conjecture: The double Lambert series

$$Y(q) = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^m q^{2nm + m}}{(1-q^{2m-1})(1+q^n)}$$

is shown to be an odd function of $q$. This had been conjectured by Andrews, Dixit, Schultz, and Yee, with significant implications for congruence properties of overpartition statistics. A new representation,

$$Y(q) = -q \frac{(q^4; q^4)_\infty^4}{(q^2; q^2)_\infty^2} \sum_{k=1}^\infty \frac{q^{2k}}{1 + q^{2k}},$$

immediately implies the desired parity. The proof leverages classical transformations of $q$-series, systematic rearrangement and evaluation of double sums, and parity analysis via explicit product formulas.

2. Confirmation of the AAB (2026) Conjecture: For any $r \in \mathbb{N}$, the coefficient of $q^{2r}$ is equated for two highly structured series:

$$[q^{2r}] \sum_{n=1}^\infty \sum_{m=1}^\infty \frac{q^{2mn}}{(1+q^{2n-1})(1-q^{2m-1})} = [q^{2r}] \sum_{n=1}^\infty \frac{(n-1)q^n}{1 + q^{2n-1}},$$

giving an analytic bridge between a double Lambert series and a single Lambert-type series with weighted coefficients. This settles a conjecture from Amdeberhan, Andrews, and Ballantine's systematic study of such series.

Analytic Techniques and Proof Structure

The proofs are based on a sequence of delicate manipulations:

• Auxiliary Series Decomposition: Multiple auxiliary double sums ($\sum_{n=1}^\infty a_n q^n / (1-q^n)$0, etc.) are defined to recast the primary series and enable recursive decomposition and parity considerations.
• Index Shifts and Series Rearrangement: Careful shifting of indices, exchange of summations, and the use of $\sum_{n=1}^\infty a_n q^n / (1-q^n)$1-Pochhammer symbols allows the condensation of multi-index sums into analytically tractable expressions.
• Application of Known Product Formulas: The authors utilize modular product expansions for series such as $\sum_{n=1}^\infty a_n q^n / (1-q^n)$2, reducing complex double sums to explicit infinite product and single-sum expressions.

A critical analytic step exploits the property that $\sum_{n=1}^\infty a_n q^n / (1-q^n)$3, i.e., a double Lambert series is invariant under $\sum_{n=1}^\infty a_n q^n / (1-q^n)$4. This, in conjunction with explicit computation and the oddness result for $\sum_{n=1}^\infty a_n q^n / (1-q^n)$5, enables the translation of conjectural combinatorial congruences into proofs via analytic $\sum_{n=1}^\infty a_n q^n / (1-q^n)$6-series identities.

Numerical Consequences and Explicit Evaluations

The identities lead to transparent analytic expressions. For example, the explicit series-form for $\sum_{n=1}^\infty a_n q^n / (1-q^n)$7 translates, via coefficient extraction and modular properties, into congruence statements about overpartition counts, such as: $\sum_{n=1}^\infty a_n q^n / (1-q^n)$8 These congruences, previously derived via more ad hoc combinatorial arguments, now follow directly from the established analytic identities.

Implications and Theoretical Significance

From a structural perspective, this work demonstrates that parity and modular phenomena in partition-theoretic identities can be realized via analytic identities among double and single Lambert series. The methods developed offer a blueprint for proving further identities involving series of this type, especially in settings where combinatorial description alone is insufficient for mod $\sum_{n=1}^\infty a_n q^n / (1-q^n)$9 congruences or for transformations between multi-dimensional and single-dimensional Lambert series.

The explicit connection between double Lambert series and $\{a_n\}$0-series product formulas may lead to progress on long-standing open problems concerning the modularity, transformation behavior, and combinatorial interpretations of generalized partition functions—an area of ongoing interest in analytic number theory and $\{a_n\}$1-series.

Practically, these analytic techniques can be applied to algorithmic coefficient extraction, generating function manipulation, and the study of mock theta functions, overpartitions, and modular objects. The rearrangement and transformation methods outlined are likely to find further application in the investigation of higher-order or multi-parameter generalizations of Lambert series and related objects.

Future Directions

Analytic approaches that unify the treatment of parity, modularity, and structural transformations in $\{a_n\}$2-series suggest fruitful lines for subsequent research:

• Systematic exploration of double and multiple Lambert series identities, particularly those corresponding to higher-rank partition-theoretic statistics or character-like decomposition.
• Investigation of modular and quasimodular transformation properties for the derived series and their connection to automorphic forms.
• Development of algorithmic tools for the automated discovery or verification of multi-sum $\{a_n\}$3-series identities using the framework established here.

Conclusion

The paper provides direct analytic proofs of two prominent conjectures involving Lambert series and double Lambert series, translating parity phenomena into explicit infinite product and single series forms. By establishing strong analytic identities and clarifying their partition-theoretic implications, this work advances the foundational understanding of the interplay between $\{a_n\}$4-series, modular forms, and combinatorial identities. Its methods are extensible and lay the groundwork for further exploration of structural phenomena in generalized $\{a_n\}$5-series and partition functions.