Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Truncated theta series from the Bailey lattice

Xiangyu Ding, Lisa Hui Sun

TL;DR

The article develops truncated theta-series identities via the Bailey lattice to obtain a truncated Jacobi triple product with odd basis, unifying and extending the Andrews–Gordon framework. It yields new truncations for Euler's pentagonal number theorem and Gauss theta-series on triangular and square numbers, with consequent partition-inequality results for pod, overpartitions, and ell-regular partitions. The work also verifies Ballantine–Merca conjectures in the 6-regular and related cases, notably proving the stronger conjecture when $R=3S$, and provides a cohesive narrative tying truncated theta series to Bailey pairs and lattice techniques. These results deepen understanding of truncated $q$-series and their combinatorial consequences, with potential further extensions through alternative Bailey hierarchies. $p(n)$, pod$(n)$, overline$p(n)$, $b_\\ell(n)$, and truncations of the Jacobi theta framework are given explicit nonnegativity implications via the derived identities.

Abstract

In 2012, Andrews and Merca obtained a truncated version of Euler's pentagonal number theorem and showed the nonnegativity related to partition functions. Meanwhile, Andrews-Merca and Guo-Zeng independently conjectured that the truncated Jacobi triple product series has nonnegative coefficients, which has been confirmed analytically and also combinatorially. In 2022, Merca proposed a stronger version for this conjecture. In this paper, by applying Agarwal, Andrews and Bressoud's Bailey lattice, we derive a truncated version for the Jacobi triple product series with odd basis which reduces to the Andrews-Gordon identity as a special instance. As consequences, we obtain new truncated forms for Euler's pentagonal number theorem, Gauss'theta series on triangular numbers and square numbers, which lead to inequalities for certain partition functions. Moreover, by considering a truncated theta series involving $\ell$-regular partitions, we confirm a conjecture proposed by Ballantine and Merca about 6-regular partitions and show that Merca's stronger conjecture on truncated Jacobi triple product series holds when $R = 3S$ for $S \geq 1.$

Truncated theta series from the Bailey lattice

TL;DR

The article develops truncated theta-series identities via the Bailey lattice to obtain a truncated Jacobi triple product with odd basis, unifying and extending the Andrews–Gordon framework. It yields new truncations for Euler's pentagonal number theorem and Gauss theta-series on triangular and square numbers, with consequent partition-inequality results for pod, overpartitions, and ell-regular partitions. The work also verifies Ballantine–Merca conjectures in the 6-regular and related cases, notably proving the stronger conjecture when , and provides a cohesive narrative tying truncated theta series to Bailey pairs and lattice techniques. These results deepen understanding of truncated -series and their combinatorial consequences, with potential further extensions through alternative Bailey hierarchies. , pod, overline, , and truncations of the Jacobi theta framework are given explicit nonnegativity implications via the derived identities.

Abstract

In 2012, Andrews and Merca obtained a truncated version of Euler's pentagonal number theorem and showed the nonnegativity related to partition functions. Meanwhile, Andrews-Merca and Guo-Zeng independently conjectured that the truncated Jacobi triple product series has nonnegative coefficients, which has been confirmed analytically and also combinatorially. In 2022, Merca proposed a stronger version for this conjecture. In this paper, by applying Agarwal, Andrews and Bressoud's Bailey lattice, we derive a truncated version for the Jacobi triple product series with odd basis which reduces to the Andrews-Gordon identity as a special instance. As consequences, we obtain new truncated forms for Euler's pentagonal number theorem, Gauss'theta series on triangular numbers and square numbers, which lead to inequalities for certain partition functions. Moreover, by considering a truncated theta series involving -regular partitions, we confirm a conjecture proposed by Ballantine and Merca about 6-regular partitions and show that Merca's stronger conjecture on truncated Jacobi triple product series holds when for
Paper Structure (6 sections, 17 theorems, 122 equations)

This paper contains 6 sections, 17 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Andrews--Gordon type truncated Jacobi triple product series
  4. The three classical truncated theta series
  5. Ballantine and Merca's conjectures
  6. Concluding remarks

Key Result

Theorem 1.2

We have where $i$, $k$, $\ell$ are nonnegative integers with $0 \leq i \leq k$ and $N_j=n_j+n_{j+1}+\cdots+n_{k}$.

Theorems & Definitions (26)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Conjecture 1.8: Ballantine and Merca ballantine20236
  • Lemma 2.1
  • Theorem 3.1
  • proof
  • ...and 16 more