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On spt-crank-type functions.(English)Zbl 1423.11174

Summary: In a recent paper [G. E. Andrews et al., Res. Number Theory 1, Paper No. 19, 25 p. (2015;Zbl 1386.11108)], Andrews, Dixit, and Yee introduced a new spt-type function \({\text{spt}}_{\omega }(n)\), which is closely related to Ramanujan’s third-order mock theta function \(\omega (q)\). Garvan and Jennings-Shaffer introduced a crank function which explains congruences for \({\text{spt}}_{\omega }(n)\). In this article, we study the asymptotic behavior of this crank function and confirm a positivity conjecture of the crank asymptotically. We also study a sign pattern of the crank and congruences for \({\text{spt}}_{\omega }(n)\).

MSC:

11P82 Analytic theory of partitions
11P83 Partitions; congruences and congruential restrictions

Citations:

Zbl 1386.11108

Cite

References:

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[2]Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partition. Bull. Am. Math. Soc. (N.S.) 18, 167-171 (1988) ·Zbl 0646.10008 ·doi:10.1090/S0273-0979-1988-15637-6
[3]Andrews, G.E., Askey, R., Roy, R.: Special Functions, Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2001)
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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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