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Extension of Bressoud's generalization of Borwein's conjecture and some exact results

Alexander Berkovich, Aritram Dhar

TL;DR

The paper extends Borwein's conjecture via a broad generalization of Bressoud's framework, formulating a generalized non-negativity conjecture for the generating function $D_{K,i}(N,M;\alpha,\beta)$ and deriving infinite hierarchies of nonnegative $q$-series identities. It introduces Conjecture $21$, demonstrates explicit nonnegativity instances (e.g., $D_{6,2}$ and $D_{9,3}$), and proves multiple positivity theorems (Theorems $23$, $25$, $26$, $27$) by applying positivity-preserving transformations for $q$-binomial coefficients, including $q\mapsto q^3$ liftings and iterative constructions. The work provides a unifying framework for positivity in partition-analytic $q$-series, yielding new nonnegative polynomials and connecting to Borwein–Bressoud-type conjectures through scalable hierarchies and corollaries. Overall, it advances the understanding of nonnegativity in $q$-series by combining combinatorial interpretations, generating functions, and transformation techniques to generate and certify broad families of nonnegative polynomials.

Abstract

In this paper, we conjecture an extension to Bressoud's 1996 generalization of Borwein's famous 1990 conjecture. We then state a few infinite hierarchies of non-negative $q$-series identities which are interesting examples of our proposed conjecture and Bressoud's generalized conjecture. Finally, using certain positivity-preserving transformations for $q$-binomial coefficients, we prove the non-negativity of the infinite families.

Extension of Bressoud's generalization of Borwein's conjecture and some exact results

TL;DR

The paper extends Borwein's conjecture via a broad generalization of Bressoud's framework, formulating a generalized non-negativity conjecture for the generating function and deriving infinite hierarchies of nonnegative -series identities. It introduces Conjecture , demonstrates explicit nonnegativity instances (e.g., and ), and proves multiple positivity theorems (Theorems , , , ) by applying positivity-preserving transformations for -binomial coefficients, including liftings and iterative constructions. The work provides a unifying framework for positivity in partition-analytic -series, yielding new nonnegative polynomials and connecting to Borwein–Bressoud-type conjectures through scalable hierarchies and corollaries. Overall, it advances the understanding of nonnegativity in -series by combining combinatorial interpretations, generating functions, and transformation techniques to generate and certify broad families of nonnegative polynomials.

Abstract

In this paper, we conjecture an extension to Bressoud's 1996 generalization of Borwein's famous 1990 conjecture. We then state a few infinite hierarchies of non-negative -series identities which are interesting examples of our proposed conjecture and Bressoud's generalized conjecture. Finally, using certain positivity-preserving transformations for -binomial coefficients, we prove the non-negativity of the infinite families.
Paper Structure (10 sections, 13 theorems, 67 equations, 2 figures)

This paper contains 10 sections, 13 theorems, 67 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Some Positivity-Preserving Transformations
  4. Proofs of some Main Results
  5. Proof of Theorem \ref{['thm23']}
  6. Proof of Theorem \ref{['thm25']}
  7. Proof of Theorem \ref{['thm26']}
  8. Proof of Theorem \ref{['thm27']}
  9. Acknowledgments
  10. Declaration

Key Result

Theorem 1.1

The generating function $D_{K,i}(N,M;\alpha,\beta)$ of partitions with at most $M$ parts, largest part not exceeding $N$, and hook differences on the $(1-\beta)$th diagonal at least $\beta-i+1$ and on the $(\alpha-1)$th diagonal at most $K-\alpha-i-1$ is given by Here the following conditions apply: $\alpha,\beta\in\mathbb{N}\cup\{0\}$, $0 < i < K$, and $\beta-i\le N-M\le K-\alpha-i$ with the add

Figures (2)

  • Figure 1: Hook differences of the partition $\pi = (5,3,1)$
  • Figure 2: Diagonals of the partition $\pi = (5,3,1)$

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1
  • Corollary 1.2
  • Remark 2
  • Remark 3
  • Conjecture 1.3
  • Conjecture 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • ...and 8 more