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

Abstract: Andrews, Dixit, Schultz, and Yee conjecture the parity of a double Lambert series. In 2026, Amdeberhan, Andrews, and Ballantine offer some ideas that are pointing in the right direction for the proof. In this paper, we complete the rest of their proof.

• The paper establishes that the double Lambert series Y(q) is odd by proving that Y(-q) = -Y(q) through analytic decomposition.
• It introduces auxiliary functions like Z(q), A(q), and B1(q) to reformulate and symmetrize the series, overcoming previous technical challenges.
• The approach leverages classical q-series identities and detailed parity-symmetrization lemmas, opening paths for further studies in q-hypergeometric series and partition theory.

Resolution of the Double Lambert Series Conjecture of Andrews–Dixit–Schultz–Yee

Introduction

The paper addresses a conjecture posed by Andrews, Dixit, Schultz, and Yee concerning the parity properties of a specific double Lambert series. Double Lambert series play an instrumental role in the analytic theory of $q$-series, partition identities, and modular-type decompositions. Prior partial progress was made by Amdeberhan, Andrews, and Ballantine, who derived alternate representations but encountered technical challenges preventing a complete resolution. This work systematically builds upon that foundation, introduces crucial auxiliary functions, and establishes the conjecture by rigorous analytic manipulations and analytical identities in $q$-series theory.

Statement of the Double Lambert Series Conjecture

At the heart of the discussion is the function

$$Y(q) := \sum_{m,n \geq 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})},$$

where $|q|<1$. Andrews et al. conjectured that $Y(q)$ is an odd function of $q$. Equivalently, $Y(-q) = -Y(q)$ for $|q|<1$. This parity property reflects a deep symmetry underlying the double sum's structure, connecting the analytic behavior of $q$-hypergeometric series with combinatorial interpretations appearing in the theory of overpartitions and mock theta functions.

Analytic Reduction and Representations

A central obstacle in earlier approaches was the intractability of classical rearrangements of $Y(q)$. The prior representation,

$q$0

while valid, proved resistant to parity analysis. The present work introduces alternative series decompositions and new auxiliary objects:

• $q$1: parametrizes a sum akin to $q$2, facilitating cancellation and parity extraction,
• $q$3, $q$4, $q$5: nested double series crafted to symmetrize the $q$6-powers and enable variable changes,
• $q$7 and $q$8: combinations of $q$9, $$Y(q) := \sum_{m,n \geq 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})},$$0, $$Y(q) := \sum_{m,n \geq 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})},$$1, $$Y(q) := \sum_{m,n \geq 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})},$$2, $$Y(q) := \sum_{m,n \geq 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})},$$3 designed to separate even and odd contributions.

A critical technical advance is re-writing $$Y(q) := \sum_{m,n \geq 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})},$$4 in terms of $$Y(q) := \sum_{m,n \geq 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})},$$5, $$Y(q) := \sum_{m,n \geq 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})},$$6, $$Y(q) := \sum_{m,n \geq 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})},$$7, and $$Y(q) := \sum_{m,n \geq 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})},$$8, and then applying analytic $$Y(q) := \sum_{m,n \geq 1} \frac{(-q)^{2mn+m}}{(1+q^m)(1-q^{2m-1})},$$9-series identities.

Main Technical Lemmas and Proof

Two lemmas are central:

1. Lemma (Parity-symmetrization):

$|q|<1$0

This exploits a sign switch in the $|q|<1$1-powers, manifest at the combinatorial level as a reflection symmetry in the underlying partitions indexed by the double sum.

2. Lemma ($|q|<1$2-series Evaluation):

$|q|<1$3

The proof here hinges on the classical $|q|<1$4-series identity:

$|q|<1$5

and the careful parameterization of the coefficients. This reduction effectively expresses the remaining difference in terms of canonical $|q|<1$6-products and a straightforward Lambert series, whose parity is manifest.

Together, these structure-preserving manipulations and reductions prove that $|q|<1$7 is indeed odd with respect to $|q|<1$8. The approach leverages intricate $|q|<1$9-hypergeometric techniques but is remarkably elementary in its execution, requiring only core $Y(q)$0-series and partition-theoretic tools beyond the initial rearrangements.

Implications

The completion of the proof of the double Lambert series conjecture resolves a parity question that links double sum analytic objects with combinatorial automorphisms. In particular, it supplies a new explicit odd function in the context of $Y(q)$1-hypergeometric and overpartition theory, enriching the landscape of identities relatable to mock theta functions and modular objects.

This formalism suggests novel avenues for the analysis of multi-parameter $Y(q)$2-series with symmetries and parity properties, and potentially, for the systematic classification of double and multiple Lambert-type series. Furthermore, as noted in the paper, the main analytic difficulty could be circumvented by elementary means if more direct representations were unearthed, offering notable potential for streamlining proofs in related settings.

Future Directions

The methodology points to several directions:

• Exploration of more elementary or combinatorial proofs for such parity properties,
• Generalization to higher-order or multivariate Lambert series,
• Analysis of the modular or mock modular nature of these double series,
• Engagement with analogous conjectures in the context of $Y(q)$3-orthogonal polynomials and partition ranks.

The explicit expression for $Y(q)$4 in terms of infinite products is of independent interest and may have consequences for the study of continued fractions, mock theta functions, and the development of $Y(q)$5-series transformation theory.

Conclusion

This work achieves a full proof of the double Lambert series conjecture of Andrews, Dixit, Schultz, and Yee by judicious analytic decomposition, auxiliary function construction, and application of classical $Y(q)$6-series identities. The result consolidates the connection of double Lambert sums to symmetry phenomena in analytic and algebraic combinatorics, and sets the stage for extended theoretical exploration of parity and modularity in $Y(q)$7-hypergeometric series.