Truncated Theta Series Related to the Jacobi Triple Product Identity

TL;DR

This work advances the theory of truncated theta-series identities tied to the Jacobi triple product by proving the first three cases of a conjecture of Merca, and by providing both analytic and combinatorial proofs of related truncated identities. It develops a rich combinatorial framework—via injections, modular diagrams, and sign-reversing involutions—to establish nonnegativity results for partition-counting sequences such as $a_{S,R,k}(n)$, $b_3(n)$, $b_6(n)$, and $Q(n)$, and ties these to truncated JTP phenomena and mex-type statistics. The paper also connects these identities to distinct 5-regular partitions through the Weierstrass addition formula, delivering new interpretations and inequalities for partition-related objects. Collectively, the results deepen understanding of how truncated theta series encode partition-structure phenomena and suggest further combinatorial routes for Jacobi-triple-product-type identities with potential applications in partition theory and modular forms.

Abstract

The work of Andrews and Merca on the truncated Euler's pentagonal number theorem led to a resurgence in research on truncated theta series identities. In particular, Yee proved a truncated version of the Jacobi Triple Product (JTP) identity. Recently, Merca conjectured a stronger form of the truncated JTP identity. In this article we prove the first three cases of the conjecture and several related truncated identities. We prove combinatorially an identity related to the JTP identity which in particular cases reduces to identities conjectured by Merca and proved analytically by Krattenthaler, Merca and Radu. Moreover, we introduce a new combinatorial interpretation for the number of distinct 5-regular partitions of n.

Theorems & Definitions (22)

• Conjecture 1
• Theorem 1.1
• Remark 1
• Theorem 1.2
• Theorem 1.3
• Remark 2
• Theorem 1.4
• proof : Proof of Theorem \ref{['main']}
• Conjecture 2
• Corollary 3.1