Bilateral truncated quintuple product

TL;DR

This work establishes bilateral truncated forms for the quintuple product identity and two well-known $q$-series consequences. By introducing sign-weighted coefficient functions and applying a key lemma together with established truncated results, the authors prove that the bilateral truncated sums have nonnegative coefficients for all positive powers of $q$. The proofs for the (3n+1) and (6n+1) cases rely on intricate decompositions and Ramanujan-type identities to control sign patterns, ensuring the desired positivity in the bilateral setting. The results yield new bilateral, nonnegative expansions and connect to combinatorial interpretations, such as overpartition triplets, enriching the theory of truncated and bilateral $q$-series identities with potential combinatorial implications.

Abstract

In this paper, we present the bilateral truncated identity of the quintuple product identity, which is a generalization of the truncated quintuple product identities given by Chan, Ho and Mao [J. Number Theory 169 (2016) 420--438]. Additionally, we provide the bilateral truncated forms of two $q$-series identities, which are well-known consequences of the quintuple product identity.

Theorems & Definitions (28)

• Theorem 1.1: chan2016truncated
• Theorem 1.2: chan2016truncated
• Theorem 1.3
• Theorem 1.4: chan2016truncated
• Theorem 1.5: chan2016truncated
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Lemma 2.1
• proof