Open Access

Partitions associated with the Ramanujan/Watson mock theta functions ω(q), ν(q)and ϕ(q)

Research in Number Theory20151:19

https://doi.org/10.1007/s40993-015-0020-8

Received: 23 March 2015

Accepted: 19 August 2015

Published: 8 December 2015

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 ω(q) (resp. ν(−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 ϕ(q). Congruences for the smallest parts partition function(s) associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler’s pentagonal number theorem are also obtained.

Keywords

Partitions Generating functions Smallest parts functions Mock theta functions Pentagonal number theorem