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The Ramanujan Journal

, Volume 45, Issue 1, pp 211–225 | Cite as

On spt-crank-type functions

  • Min-Joo Jang
  • Byungchan KimEmail author
Article
  • 249 Downloads

Abstract

In a recent paper, Andrews, Dixit, and Yee introduced a new spt-type function \({\mathrm {spt}}_{\omega }(n)\), 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 \({\mathrm {spt}}_{\omega }(n)\). 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 \({\mathrm {spt}}_{\omega }(n)\).

Keywords

Spt function Spt-crank function Asymptotic formula Partial theta function Congruences 

Mathematics Subject Classification

11P82 

Notes

Acknowledgments

This paper will be a part of the first author’s PhD thesis. The authors thank Kathrin Bringmann, Michael Woodbury, and the referee for their valuable comments on an earlier version of this paper.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of CologneCologneGermany
  2. 2.School of Liberal ArtsSeoul National University of Science and TechnologySeoulKorea

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