Skip to main content
SpringerLink
Log in

A Ramanujan-type congruence modulo 5 for 4-colored generalized Frobenius partitions

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

In his 1984 AMS Memoir, Andrews introduced the \(k\)-colored generalized Frobenius partition function \(c\phi _k(n)\) which denotes the number of generalized Frobenius partitions of \(n\) with \(k\) colors. Recently, Baruah and Sarmah, Lin, and Sellers established several Ramanujan-type congruences for \(c\phi _4(n)\). In this paper, employing some theta identities due to Ramanujan, the \((p, k)\)-parametrization of theta functions given by Alaca, Alaca, and Williams, and some results of Baruah and Sarmah, we prove that \(c\phi _4(20n+11)\equiv 0\ (\mathrm{mod}\ 5)\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alaca, A., Alaca, S., Williams, K.S.: On the two-dimensional theta functions of the Borweins. Acta Arith. 124, 177–195 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alaca, S., Williams, K.S.: The number of representations of a positive integer by certain octonary quadratic forms. Funct. Approx. Comment. Math. 43, 45–54 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrews, G.E.: Generalized Frobenius Partitions, Memoirs of the American Mathematical Society, vol. 301. American Mathematical Society, Providence (1984)

    Google Scholar 

  4. Baruah, N.D., Sarmah, B.K.: Congruences for generalized Frobenius partitions with 4 colors. Discret. Math. 311, 1892–1902 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baruah, N.D., Sarmah, B.K.: Generalized Frobenius partitions with 6 colors. Ramanujan J. doi:10.1007/s11139-014-9595-2

  6. Berndt, B.C.: Ramanujan’s Notebooks, vol. III. Springer, New York (1991)

    Book  MATH  Google Scholar 

  7. Eichhorn, D., Sellers, J.A.: Computational proofs of congruences for 2-colored Frobenius partitions. Int. J. Math. Math. Sci. 29, 333–340 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Garvan, F.G., Sellers, J.A.: Congruences for generalized Frobenius partitions with an arbitrarily large number of colors, Integers 14, Article A7 (2014)

  9. Hirschhorn, M.D., Sellers, J.A.: Elementary proofs of parity results for \(5\)-regular partitions. Bull. Aust. Math. Soc. 81, 58–63 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kolitsch, L.W.: An extension of a congruence by Andrews for generalized Frobenius partitions. J. Combin. Theory Ser. A 45, 31–39 (1987)

  11. Kolitsch, L.W.: A congruence for generalized Frobenius partitions with 3-colors modulo powers of 3. In: Analytic Number Theory: Proceedings of a Conference in Honor of Paul T. Bateman, pp. 343–348, Birkhäuser, Boston (1990)

  12. Lin, B.L.S.: New Ramanujan type congruence modulo 7 for 4-colored generalized Frobenius partitions. Int. J. Number Theory 10, 637–639 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lovejoy, J.: Ramanujan like congruences for three colored Frobenius partitions. J. Number Theory 85, 283–290 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ono, K.: Congruences for Frobenius partitions. J. Number Theory 57, 170–180 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Paule, P., Radu, C.S.: The Andrews–Sellers family of partition congruences. Adv. Math. 230, 819–838 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sellers, J.A.: Congruences involving generalized Frobenius partitions. Int. J. Math. Math. Sci. 16, 413–415 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sellers, J.A.: New congruences for generalized Frobenius partitions with 2 or 3 colors. Discret. Math. 131, 367–374 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sellers, J.A.: An unexpected congruence modulo 5 for 4-colored generalized Frobenius partitions, J. Indian Math. Soc., Special Volume to Commemorate the 125th Birth Anniversary of Srinivasa Ramanujan, pp. 97–103 (2013)

  19. Williams, K.S.: Fourier series of a class of eta quotients. Int. J. Number Theory 8, 993–1004 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We wish to thank the referees for their helpful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Jiangsu University, Zhenjiang, 212013, Jiangsu, People’s Republic of China

    Ernest X. W. Xia

Authors
  1. Ernest X. W. Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ernest X. W. Xia.

Additional information

This work was supported by the National Natural Science Foundation of China (11201188).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xia, E.X.W. A Ramanujan-type congruence modulo 5 for 4-colored generalized Frobenius partitions. Ramanujan J 39, 567–576 (2016). https://doi.org/10.1007/s11139-014-9642-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-014-9642-z

Keywords

Mathematics Subject Classification

Search

Navigation