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The $p$-Dissection of a Product of Quintuple Products

Taylor Daniels, Timothy Huber, James McLaughlin, Dongxi Ye

Abstract

Let $p \equiv 1 \pmod{4}$ be prime, let $m$ and $n$ be integers such that $p=m^2+n^2$, and let $b$ be a positive integer. Let $Q(z,q) = (z,q/z,q;q)_{\infty}(qz^2,q/z^2;q^2)_{\infty}$ denote the product appearing in the quintuple product identity. We derive explicit formulae for the $p$-dissection of $Q(q^{bm},q^p)Q(q^{bn},q^p)$, and determine sign patterns in length-$p$ arithmetic progressions of the Taylor series coefficients of the associated quotient $Q(q^{bm},q^{p})Q(q^{bn},q^p)/(q^p;q^p)_{\infty}^2$. Some combinatorial applications of the $p$-dissection formulae are also given.

The $p$-Dissection of a Product of Quintuple Products

Abstract

Let be prime, let and be integers such that , and let be a positive integer. Let denote the product appearing in the quintuple product identity. We derive explicit formulae for the -dissection of , and determine sign patterns in length- arithmetic progressions of the Taylor series coefficients of the associated quotient . Some combinatorial applications of the -dissection formulae are also given.
Paper Structure (9 sections, 19 theorems, 136 equations)

This paper contains 9 sections, 19 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A key dissection lemma
  4. The case p≡ 1(mod12)
  5. Sign patterns when p≡1(mod 12)
  6. The case p≡5(mod 12)
  7. Sign patterns when p≡5(mod 12)
  8. Combinatorial interpretations
  9. Concluding remarks

Key Result

Theorem 1

Fix a prime $p \equiv 1 \,\,(\mathrm{mod}\,4)$, write $p = m^2 + n^2$, let $b > 0$, and define $(a_{t})_{t \in \mathbb{Z}}$ as in eq:atDefin. In addition, let $w$ satisfy If $p \equiv 1 \,\,(\mathrm{mod}\,12)$, then and if $p \equiv 5 \,\,(\mathrm{mod}\,12)$, then

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Corollary 1
  • proof
  • Lemma 2: LY09
  • Definition 1
  • Definition 2
  • Corollary 2
  • ...and 31 more