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The Ramanujan Journal

, Volume 43, Issue 2, pp 347–357 | Cite as

Congruences related to the Ramanujan/Watson mock theta functions \(\omega (q)\) and \(\nu (q)\)

  • George E. Andrews
  • Donny Passary
  • James A. Sellers
  • Ae Ja Yee
Article

Abstract

Recently, Andrews, Dixit, and Yee introduced partition functions associated with the Ramanujan/Watson mock theta functions \(\omega (q)\) and \(\nu (q)\). In this paper, we study arithmetic properties of the partition functions. Based on one of the results of Andrews, Dixit, and Yee, mod 2 congruences are obtained. In addition, infinite families of mod 4 and mod 8 congruences are presented. Lastly, an elementary proof of the first explicit examples of congruences for \(\omega (q)\) given by Waldherr is presented.

Keywords

Partition congruences Generating function Mock theta functions 

Mathematics Subject Classification

Primary 11P81 11P83 

Notes

Acknowledgments

The authors are grateful to the referee for very valuable comments and suggestions.

References

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    Garthwaite, S.A., Penniston, D.: \(p\)-Adic properties of Maass forms arising from theta series. Math. Res. Lett. 15, 459–470 (2008)MathSciNetCrossRefMATHGoogle Scholar
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    Waldherr, M.: On certain explicit congruences for mock theta functions. Proc. Am. Math. Soc. 139, 865–879 (2011)MathSciNetCrossRefMATHGoogle Scholar
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    Watson, G.N.: The final problem: an account of the mock theta functions. J. Lond. Math. Soc. 11, 55–80 (1936)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • George E. Andrews
    • 1
  • Donny Passary
    • 1
  • James A. Sellers
    • 1
  • Ae Ja Yee
    • 1
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

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