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Congruences for the coefficients of the Gordon and McIntosh mock theta function $ξ(q)$

Robson da Silva, James A. Sellers

Abstract

Recently Gordon and McIntosh introduced the third order mock theta function $ξ(q)$ defined by $$ ξ(q)=1+2\sum_{n=1}^{\infty}\frac{q^{6n^2-6n+1}}{(q;q^6)_{n}(q^5;q^6)_{n}}. $$ Our goal in this paper is to study arithmetic properties of the coefficients of this function. We present a number of such properties, including several infinite families of Ramanujan--like congruences.

Congruences for the coefficients of the Gordon and McIntosh mock theta function $ξ(q)$

Abstract

Recently Gordon and McIntosh introduced the third order mock theta function defined by Our goal in this paper is to study arithmetic properties of the coefficients of this function. We present a number of such properties, including several infinite families of Ramanujan--like congruences.

Paper Structure

This paper contains 6 sections, 21 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dissections for $p_{\xi}(n)$
  4. Arithmetic properties of $p_{\xi}(n)$
  5. Additional congruences
  6. Concluding remarks

Key Result

Lemma 2.1

The following 2-dissection identities hold.

Theorems & Definitions (44)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • ...and 34 more