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The Ramanujan Journal

, Volume 40, Issue 3, pp 649–668 | Cite as

New congruences for \(\ell \)-regular partitions for \(\ell \in \{5,6,7,49\}\)

  • Zakir Ahmed
  • Nayandeep Deka Baruah
Article

Abstract

We find several new congruences for \(\ell \)-regular partitions for \(\ell \in \{5,6,7,49\}\) and also find alternative proofs of the congruences for 10- and 20-regular partitions which were proved earlier by Carlson and Webb (Ramanujan J 33:329–337, 2014) by using the theory of modular forms. We use certain p-dissections of \((q;q)_{\infty }\), \(\psi (q)\), \((q;q)_{\infty }^3\) and \(\psi (q^2)(q;q)_{\infty }^2\).

Keywords

\(\ell \)-Regular partition p-Dissection Congruence 

Mathematics Subject Classification

Primary 11P83 Secondary 05A15 05A17 

Notes

Acknowledgments

The authors thank the anonymous referee for his/her insightful comments.

References

  1. 1.
    Baruah, N.D., Das, K.: Parity results for 7-regular and 23-regular partitions. Int. J. Number Theory 11, 2221–2238 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Berndt, B.C.: Ramanujan’s Notebooks. Part III. Springer, New York (1991)CrossRefMATHGoogle Scholar
  3. 3.
    Berndt, B.C.: Number Theory in the Spirit of Ramanujan. American Mathematical Society, Providence (2006)CrossRefMATHGoogle Scholar
  4. 4.
    Carlson, R., Webb, J.J.: Infinite families of congruences for \(k\)-regular partitions. Ramanujan J. 33, 329–337 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cui, S.P., Gu, N.S.S.: Arithmetic properties of \(\ell \)-regular partitions. Adv. Appl. Math. 51, 507–523 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dandurand, B., Penniston, D.: \(\ell \)-Divisibility of \(\ell \)-regular partition functions. Ramanujan J. 19, 63–70 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Furcy, D., Penniston, D.: Congruences for \(\ell \)-regular partitions modulo \(3\). Ramanujan J. 27, 101–108 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Garvan, F.G.: A simple proof of Watson’s partition congruences for powers of \(7\). J. Aust. Math. Soc. A 36, 316–334 (1984)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hou, Q.-H., Sun, L.H., Zhang, L.: Quadratic forms and congruences for \(\ell \)-regular partitions modulo 3, 5 and 7. Adv. Appl. Math. 70, 32–44 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lin, B.L.S.: An infinite family of congruences modulo 3-regular bipartitions. Ramanujan J. 2014. doi: 10.1007/s11139-014-9610-7
  11. 11.
    Lin, B.L.S.: Arithmetic of the \(7\)-regular bipartition function modulo \(3\). Ramanujan J. 37, 469–478 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Webb, J.J.: Arithmetic of the 13-regular partition function modulo 3. Ramanujan J. 25, 49–56 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTezpur UniversitySonitpurIndia

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