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Multiple Rogers--Ramanujan type identities for torus links

Shane Chern

Abstract

In this paper, we establish simple $k$-fold summation expressions for the Quot and motivic Cohen--Lenstra zeta functions associated with the $(2,2k)$ torus links. Such expressions lead us to some multiple Rogers--Ramanujan type identities and their finitizations, thereby confirming a conjecture of Huang and Jiang. Several other properties of the two zeta functions will be examined as well.

Multiple Rogers--Ramanujan type identities for torus links

Abstract

In this paper, we establish simple -fold summation expressions for the Quot and motivic Cohen--Lenstra zeta functions associated with the torus links. Such expressions lead us to some multiple Rogers--Ramanujan type identities and their finitizations, thereby confirming a conjecture of Huang and Jiang. Several other properties of the two zeta functions will be examined as well.

Paper Structure

This paper contains 11 sections, 28 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. $q$-Series prerequisites
  3. Iteration seed toward Theorem \ref{['th:multi-sum-infty']}
  4. Theorem \ref{['th:S-m-infty']} and its finitization
  5. Reformulating $\mathcal{Z}_k(N;t,q)$
  6. A semi-truncation
  7. $q$-Lebesgue identity
  8. Toward the $\mathrm{A}_1$-type sum in Theorem \ref{['th:Z-expression']}
  9. Theorem \ref{['th:S-m-infty']} revisited
  10. Finitization of Theorems \ref{['th:S-m-(-1)-infty']} and \ref{['th:strange-A2-sum']}
  11. Huang and Jiang's reflection formula

Key Result

Theorem 1.1

For any positive integer $k$,

Theorems & Definitions (47)

  • Theorem 1.1: Huang--Jiang
  • Conjecture 1.1: Huang--Jiang
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Conjecture 1.2: Huang--Jiang, Nonnegativity Conjecture
  • ...and 37 more