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On spt-crank-type functions

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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 \({\mathrm {spt}}_{\omega }(n)\).

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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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Authors and Affiliations

  1. Mathematical Institute, University of Cologne, Weyertal 86-90, 50931, Cologne, Germany

    Min-Joo Jang

  2. School of Liberal Arts, Seoul National University of Science and Technology, 232 Gongneung-ro, Nowon-gu, Seoul, 01811, Korea

    Byungchan Kim

Authors
  1. Min-Joo Jang
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  2. Byungchan Kim
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Correspondence to Byungchan Kim.

Additional information

Byungchan Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2061326).

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Jang, MJ., Kim, B. On spt-crank-type functions. Ramanujan J 45, 211–225 (2018). https://doi.org/10.1007/s11139-016-9838-5

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  • DOI: https://doi.org/10.1007/s11139-016-9838-5

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