Proof of a conjecture of Andrews and Bachraoui on a Hecke sum

Technical Summary of "Proof of a Conjecture of Andrews and Bachraoui on a Hecke Sum"

Problem Statement
This paper addresses a conjecture proposed by Andrews and Bachraoui regarding the generating function for a specific class of two-color partitions, denoted $C(k, n)$. These partitions involve restrictions on the smallest part (must be odd and blue) and constraints on even parts (distinctness and a lower bound relative to the smallest part). As $k \to \infty$, the generating function $C_k(q)$ converges to a limit function $C(q) = qD(q)S(q)$, where $S(q)$ is a specific $q$-series.

Andrews and Bachraoui conjectured that the even part of the generating function, obtained by applying a sieving operator $S_{2,0}$ to $(q^4; q^4)_\infty S(q)$, is equal to a specific Hecke-type double sum. Specifically, Conjecture 1.1 posits the identity:
$$\left( (q^4; q^4)_\infty S(q) \right) \big|_{S_{2,0}} = \sum_{n \ge 0} (-1)^n q^{6n^2+4n} (1 + q^{4n+2}) \sum_{|j| \le n} (-1)^j q^{-2j^2}$$
The primary objective of this work is to provide a rigorous proof of this identity.

Methodology
The proof strategy relies on the theory of indefinite theta functions and modular forms, specifically utilizing the framework developed by Zwegers. The authors employ the following methodological steps:

1. Completion to Real-Analytic Modular Forms: Both sides of the conjectured identity are "completed" by adding non-holomorphic terms to form real-analytic modular forms.

   • The right-hand side (the Hecke sum) is realized as the holomorphic part of an indefinite theta function $\vartheta_{a,b}(\tau)$ constructed by Zwegers. A non-holomorphic correction term $H^-(\tau)$ is explicitly defined to complete the function $\hat{H}(\tau)$.
   • The left-hand side is rewritten using known relations between $S(q)$ and Ramanujan's mock theta functions $\omega(q)$ and $B(q)$. Through the use of sieving operators and transformation properties of the Dedekind eta function and theta functions, the left-hand side is expressed in terms of mock theta functions and their Zwegers completions ($\hat{F}_j$). A corresponding non-holomorphic term $A^-(\tau)$ is constructed to form the completed function $\hat{A}(\tau)$.

2. Modular Transformation Analysis: The authors analyze the modular transformation properties of the completed functions $\hat{H}(\tau)$ and $\hat{A}(\tau)$.

   • They demonstrate that both functions satisfy specific transformation laws under the action of the congruence subgroup $\Gamma_0(2)$, particularly under the generators $\tau \mapsto \tau + 1$ and $\tau \mapsto \frac{\tau}{2\tau+1}$.
   • The transformation properties of the indefinite theta function $\hat{H}$ are derived using Zwegers' general theorems on indefinite theta functions, including elliptic and modular transformations.
   • The transformations for $\hat{A}$ are derived by combining the known modular properties of the eta function, the theta function $\Theta$, and the vector-valued modular form $\hat{F}$ associated with the mock theta functions.

3. Application of Sturm's Theorem: The difference between the two completed functions, $\hat{M}(\tau) = \hat{H}(\tau) - \hat{A}(\tau)$, is shown to be a holomorphic modular form of weight $1/2$ (after appropriate scaling) on $\Gamma_0(2)$.

   • The authors verify that $\hat{M}(\tau)$ has at most polynomial growth at the cusps of $\Gamma_0(2)$ (specifically $i\infty$ and $0$).
   • By Sturm's theorem, to prove that a modular form is identically zero, it suffices to show that a finite number of its Fourier coefficients vanish. The number of required coefficients is determined by the weight and the index of the congruence subgroup.
   • A direct computation confirms that the necessary initial Fourier coefficients of $\hat{M}(\tau)$ are zero.

Key Contributions and Results

• Proof of Conjecture 1.1: The paper establishes the validity of the Andrews-Bachraoui conjecture, confirming the equality between the sieved generating function of the two-color partitions and the Hecke-type double sum.
• Explicit Completion: The authors provide explicit constructions of the non-holomorphic completions for both the Hecke sum and the sieved partition generating function, demonstrating that they fit into the same vector-valued modular framework.
• Modular Properties: The work details the specific modular transformation properties of these completed functions under $\Gamma_0(2)$, linking combinatorial $q$-series to the theory of harmonic Maass forms and indefinite theta functions.

Significance
The paper claims significance primarily in resolving a specific open problem in the theory of partitions and $q$-series. By proving the conjecture, the authors connect a combinatorial object (two-color partitions with specific constraints) to a Hecke-type double sum, a structure often associated with mock modular forms. The methodology highlights the utility of Zwegers' theory of indefinite theta functions in proving identities involving mock theta functions and sieved series. The authors note that their functions are components of vector-valued harmonic Maass forms, suggesting a deeper structural relationship between these combinatorial generating functions and the broader theory of modular forms.

Future Directions
The paper concludes by outlining several open questions for future research, including:

• Investigating whether the holomorphic parts of other components of the associated vector-valued harmonic Maass forms possess combinatorial interpretations.
• Extending the study of sieved generating functions to odd residue classes.
• Exploring extensions of these results to other mock theta functions.
• Seeking a direct proof of the conjecture, potentially utilizing Bailey pairs, rather than the modular completion approach used here.