Truncated theta series related to the Jacobi triple product identity
type: Item
instance of: scholarly article
title: Truncated theta series related to the Jacobi triple product identity
author: Cristina Ballantine
author: Brooke Feigon
published in: Discrete Mathematics
publication date: 2024-12-02T00:00:00Z
review text: The collaboration between \textit{G. E. Andrews} and \textit{M. Merca} [J. Comb. Theory, Ser. A 119, No. 8, 1639--1643 (2012; Zbl 1246.05014)] on the truncated Euler's pentagonal number theorem has invigorated the field of truncated theta series identities, sparking a wave of innovative research. Building upon this momentum, the current article makes significant strides in advancing our understanding of truncated identities and their combinatorial underpinnings.\N\NOne of the standout achievements of this paper is the successful proof of the first three cases of Merca's conjecture on the truncated Jacobi Triple Product (JTP) identity [the reviewer, Exp. Math. 31, No. 2, 606--610 (2022; Zbl 1512.11031)]. This accomplishment not only validates the rewiever's stronger form of the conjecture but also lays a robust foundation for future explorations into higher cases and related conjectures. Additionally, the authors adeptly tackle several related truncated identities, demonstrating a comprehensive approach to the problem.\N\NThe combinatorial proof of an identity related to the JTP identity is particularly noteworthy. By providing a combinatorial perspective, the authors offer deeper insights that complement the analytical proofs previously established by \textit{C. Krattenthaler} et al. [Springer Proc. Math. Stat. 373, 193--236 (2021; Zbl 1499.11189)]. This dual approach enriches the theoretical framework and enhances the overall understanding of the identities in question.\N\NMoreover, the introduction of a new combinatorial interpretation for the number of distinct \(5\)-regular partitions of \(n\) is a significant contribution. This novel interpretation not only broadens the scope of partition theory but also opens up new avenues for research into regular partitions and their applications.\N\NThe methodology employed throughout the paper is both rigorous and elegant. The clear and logical presentation of complex ideas makes the work accessible to a broad audience, w…
reviewed by: Mircea Merca
Mathematics Subject Classification ID: 11P81
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 11P84
zbMATH DE Number: 7952118
zbMATH Keywords: partitions
zbMATH Keywords: theta series
zbMATH Keywords: truncated series
zbMATH Keywords: combinatorial injection
MaRDI profile type: Publication
cites work: Q4218535
cites work: Two theorems of Gauss and allied identities proved arithmetically
cites work: The Fifth and Seventh Order Mock Theta Functions
cites work: Q3156941
cites work: The truncated pentagonal number theorem
cites work: Q5109525
cites work: Hook length biases and general linear partition inequalities
cites work: Almost 3-regular overpartitions
cites work: 6-regular partitions: new combinatorial properties, congruences, and linear inequalities
cites work: Q4405657
cites work: Two truncated identities of Gauss
cites work: Combinatorial proof of a theorem on partitions into an even or odd number of parts
cites work: A combinatorial proof of Jacobi's triple product identity
cites work: Infinite product formulae for generating functions for sequences of squares
cites work: Truncated Theta Series and Rogers-Ramanujan Functions
cites work: On Two Truncated Quintuple Series Theorems
cites work: Linear inequalities concerning partitions into distinct parts
cites work: Rank partition functions and truncated theta identities
cites work: A Short Proof of an Identity of Euler
cites work: Two theorems of Gauss
cites work: Q4443440
cites work: Truncated Jacobi triple product series
cites work: Partial-sum analogues of the Rogers-Ramanujan identities
cites work: Truncated sums for the partition function and a problem of Merca
cites work: Proof of a conjecture of Ballantine and Merca on truncated sums of 6-regular partitions
cites work: A truncated Jacobi triple product theorem
DOI: 10.1016/J.DISC.2024.114319
description: scientific article; zbMATH DE number 7952118