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Congruences for partition functions related to mock theta functions

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Abstract

Partitions associated with mock theta functions have received a great deal of attention in the literature. Recently, Choi and Kim derived several partition identities from the third- and sixth-order mock theta functions. In addition, three Ramanujan-type congruences were established by them. In this paper, we present some new congruences for these partition functions.

Keywords

Partition t-Core partition Cubic partition Mock theta function Ramanujan-type congruence 

Mathematics Subject Classification

11P83 05A17 

Notes

Acknowledgements

We would like to thank the referee and editor for helpful comments.

References

  1. 1.
    Alaca, Ş., Williams, K.S.: The number of representations of a positive integer by certain octonary quadratic forms. Funct. Approx. Comment. Math 43(part 1), 45–54 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andrews, G.E.: Partitions with short sequences and mock theta functions. Proc. Natl. Acad. Sci. USA 102(13), 4666–4671 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Andrews, G.E., Garvan, F.G.: Ramanujan’s “lost” notebook. VI. The mock theta conjectures. Adv. Math. 73(2), 242–255 (1989)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Andrews, G.E., Dixit, A., Yee, A.J.: Partitions associated with the Ramanujan/Watson mock theta functions \(\omega (q)\), \(\nu (q)\) and \(\phi (q)\). Res. Number Theory1, 19 (2015)Google Scholar
  5. 5.
    Andrews, G.E., Dixit, A., Schultz, D., Yee, A.J.: Overpartitions related to the mock theta function $\omega (q)$. Preprint (2016). Available at arXiv:1603.04352 Google Scholar
  6. 6.
    Andrews, G.E., Passary, D., Seller, J., Yee, A.J.: Congruences related to the Ramanujan/Watson mock theta functions $\omega (q)$ and $\nu (q)$. Ramanujan J. 43(2), 347–357 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Baruah, N.D., Ojah, K.K.: Some congruences deducible from Ramanujan’s cubic continued fraction. Int. J. Number Theory 7(5), 1331–1343 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Berndt, B.C.: Ramanujan’s Notebooks. Part III, p. xiv+510. Springer, New York (1991)CrossRefMATHGoogle Scholar
  9. 9.
    Berndt, B.C., Rankin, R.A.: Ramanujan: Letters and Commentary. History of Mathematics Series, vol. 9. American Mathematical Society/London Mathematical Society, Providence, RI/London (1995)CrossRefMATHGoogle Scholar
  10. 10.
    Blecher, A.: Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal. Util. Math. 88, 223–235 (2012)MathSciNetMATHGoogle Scholar
  11. 11.
    Chan, H.-C.: Ramanujan’s cubic continued fraction and an analog of his “most beautiful identity”. Int. J. Number Theory 6(3), 673–680 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chan, H.-C.: Ramanujan’s cubic continued fraction and Ramanujan type congruences for a certain partition function. Int. J. Number Theory 6(4), 819–834 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chern, S.: Congruences for $1$-shell totally symmetric plane partitions. Integers 17(A21), 7 (2017)MathSciNetGoogle Scholar
  14. 14.
    Choi, Y.-S.: The basic bilateral hypergeometric series and the mock theta functions. Ramanujan J. 24(3), 345–386 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Choi, Y.-S., Kim, B.: Partition identities from third and sixth order mock theta functions. Eur. J. Combin. 33(8), 1739–1754 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cui, S.-P., Gu, N.S.S.: Arithmetic properties of $\ell $-regular partitions. Adv. Appl. Math. 51(4), 507–523 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fine, N.J.: Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, vol. 27. American Mathematical Society, Providence, RI (1988)CrossRefGoogle Scholar
  18. 18.
    Garvan, F., Kim, D., Stanton, D.: Cranks and $t$-cores. Invent. Math. 101(1), 1–17 (1990)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications, vol. 96, 2nd edn. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  20. 20.
    Hirschhorn, M.D., Sellers, J.A.: Arithmetic properties of $1$-shell totally symmetric plane partitions. Bull. Aust. Math. Soc. 89(3), 473–478 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kim, B.: An analog of crank for a certain kind of partition function arising from the cubic continued fraction. Acta Arith. 148(1), 1–19 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Radu, S., Sellers, J.A.: Congruence properties modulo $5$ and $7$ for the ${{\rm pod}}$ function. Int. J. Number Theory 7(8), 2249–2259 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Shen, L.-C.: On the modular equations of degree 3. Proc. Am. Math. Soc. 122(4), 1101–1114 (1994)MathSciNetMATHGoogle Scholar
  24. 24.
    Wang, L.: Arithmetic identities and congruences for partition triples with 3-cores. Int. J. Number Theory 12(4), 995–1010 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Xia, E.X.W.: A new congruence modulo $25$ for $1$-shell totally symmetric plane partitions. Bull. Aust. Math. Soc. 91(1), 41–46 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Xia, E.X.W., Yao, O.X.M.: Analogues of Ramanujan’s partition identities. Ramanujan J. 31(3), 373–396 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Yao, O.X.M.: New infinite families of congruences modulo $4$ and $8$ for $1$-shell totally symmetric plane partitions. Bull. Aust. Math. Soc. 90(1), 37–46 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  2. 2.Center for Combinatorics, LPMCNankai UniversityTianjinPeople’s Republic of China

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