Partitions associated with the Ramanujan/Watson mock theta functions $\omega(q), \nu(q)$ and $\phi(q)$

TL;DR

This paper links specific partition generating functions with third order mock theta functions and explores their properties, including congruences and analogues of Euler's pentagonal theorem.

Contribution

It establishes new connections between partition functions with restrictions and Ramanujan/Watson mock theta functions, providing novel interpretations and congruence results.

Findings

Generating functions for certain partitions equal third order mock theta functions.

Derived congruences for smallest parts functions of these partitions.

Presented analogues of Euler's pentagonal theorem in a partition context.

Abstract

The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function $\omega(q)$ (resp. $\nu(-q)$). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function $\phi(q)$. Congruences for the smallest parts functions associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler's pentagonal theorem are also obtained.