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.
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Acknowledgements
The authors sincerely thank the referee for his/her helpful suggestions and comments. The second author was funded in part by the grant NSF-DMS 1112656 of Professor Victor H. Moll of Tulane University, whom he sincerely thanks for this support. The third author was partially supported by a grant (#280903) from the Simons Foundation.
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Andrews, G.E., Dixit, A. & Yee, A.J. Partitions associated with the Ramanujan/Watson mock theta functions ω(q), ν(q)and ϕ(q). Res. number theory 1, 19 (2015). https://doi.org/10.1007/s40993-015-0020-8
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DOI: https://doi.org/10.1007/s40993-015-0020-8