A proof of the mod $4$ unimodal sequence conjectures and related mock theta functions

TL;DR

This paper proves several conjectures related to mod 4 congruences for coefficients of generating functions of unimodal sequences and mock theta functions, advancing understanding of their arithmetic properties.

Contribution

It provides the first proof of multiple mod 4 unimodal sequence conjectures and related mock theta function congruences using new identities involving quadratic forms.

Findings

Proved all mod 4 conjectures for unimodal sequences and mock theta functions.

Established new Hecke-Rogers identities for quadratic forms and Hurwitz class numbers.

Extended results to the Andrews spt-function and related mock theta functions.

Abstract

In 2012 Bryson, Ono, Pitman and Rhoades showed how the generating functions for certain strongly unimodal sequences are related to quantum modular and mock modular forms. They proved some parity results and conjectured some mod 4 congruences for the coefficients of these generating functions. In 2016 Kim, Lim and Lovejoy obtained similar results for odd-balanced unimodal sequences and made similar mod 4 conjectures. We prove all of these mod 4 conjectures and similar congruences for the Andrews spt-function and related mock theta functions. Our method of proof involves new Hecke-Rogers type identities for indefinite binary quadratic forms and the Hurwitz class number.