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.
Highlights
1 Introduction Partition-theoretic interpretations of various results involving mock theta functions have been the subject of intense study for many decades
The first author and Garvan [14] reduced the proofs of ten identities for the fifth order mock theta functions given in Ramanujan’s Lost Notebook ([29] p. 18–20) to proving two conjectures based on the rank of a partition, and on the number of partitions of an integer with unique smallest part and all other parts less than or equal to the double of the smallest part
The series given in the theorem is clearly the generating function for partitions in which each odd part is less than twice the smallest part
Summary
Partition-theoretic interpretations of various results involving mock theta functions have been the subject of intense study for many decades. If we put an additional restriction that the parts be distinct, the generating function of such partitions is ν(−q) (see Theorem 4.1), where ν(q) is another third order mock theta function defined by, Let pν(n) denote the number of partitions of n into distinct parts in which each odd part is less than twice the smallest part. The series given in the theorem is clearly the generating function for partitions in which each odd part is less than twice the smallest part. We obtain a representation corresponding to Theorem 3.1 for the generating function of partitions in which each even part is at most twice the smallest part. The result corresponding to Theorem 4.1 for partitions into distinct parts where each even part is at most twice the smallest part is presented below. − 1, where in the last step we used the Jacobi triple product identity ([4] p. 21, Theorem 2.8)
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