Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a formula of the $q$-series $_{2k+4}φ_{2k+3}$ and its applications

George E. Andrews, Mohamed El Bachraoui

Abstract

In this paper we apply a formula of the very-well poised $_{2k+4}φ_{2k+3}$ to write a $k$-tuple sum of $q$-series as a linear combination of terms wherein each term is a product of expressions of the form $\frac{1}{(qy, qy^{-1};q)_\infty}$. As an application, we shall express a variety of sums and double sums of $q$-series as linear combinations of infinite products. Our formulas are motivated by their connection to overpartition pairs.

On a formula of the $q$-series $_{2k+4}φ_{2k+3}$ and its applications

Abstract

In this paper we apply a formula of the very-well poised to write a -tuple sum of -series as a linear combination of terms wherein each term is a product of expressions of the form . As an application, we shall express a variety of sums and double sums of -series as linear combinations of infinite products. Our formulas are motivated by their connection to overpartition pairs.

Paper Structure

This paper contains 9 sections, 9 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. The results
  3. Proof of Theorem \ref{["thm A'-1"]}
  4. Proof of Theorem \ref{["thm A'-2"]}
  5. Proof of Theorem \ref{['thm DS-1']}
  6. Proof of Theorem \ref{['thm DS-2']} and Theorem \ref{['thm DS-3']}
  7. Proof of Theorem \ref{["thm B'"]} and Theorem \ref{['thm DS-4']}
  8. Proof of Theorem \ref{['thm general']}
  9. Concluding remarks

Key Result

Theorem 1

Let $k\in\mathbb{N}$ and $N\in\mathbb{N}_0$. Then there holds where $A_{1,1}(b_1)=1$ and

Theorems & Definitions (12)

  • Theorem 1
  • Corollary 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • ...and 2 more