The Ramanujan Journal

, Volume 46, Issue 1, pp 19–27 | Cite as

On \((\ell , m)\)-regular partitions with distinct parts

Article

Abstract

Let \(a_{\ell ,m}(n)\) denote the number of \((\ell ,m)\)-regular partitions of a positive integer n into distinct parts, where \(\ell \) and m are relatively primes. In this paper, we establish several infinite families of congruences modulo 2 for \(a_{3,5}(n)\). For example,
$$\begin{aligned} a_{3, 5}\left(2^{6\alpha +4}5^{2\beta }n+\frac{ 2^{6\alpha +3}5^{2\beta +1}-1}{3}\right) \equiv 0 , \end{aligned}$$
where \(\alpha , \beta \ge 0\).

Keywords

Partition identities Theta-functions Partition congruences Regular partition 

Mathematics Subject Classification

11P83 05A17 

Notes

Acknowledgements

The authors are thankful to the referee for useful suggestions which improved the quality of the paper.

References

  1. 1.
    Baruah, N.D., Berndt, B.C.: Partition identities and Ramanujan’s modular equations. J. Comb. Theory Ser. A 114, 1024–1045 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)CrossRefMATHGoogle Scholar
  3. 3.
    Calkin, N., Drake, N., James, K., Law, S., Lee, P., Penniston, D., Radder, J.: Divisibility properties of the \(5\)-regular and \(13\)-regular partition functions. Integers 8, A60 (2008)MathSciNetMATHGoogle Scholar
  4. 4.
    Cui, S.P., Gu, N.S.S.: Arithmetic properties of \(l\)-regular partitions. Adv. Appl. Math. 51, 507–523 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hirschhorn, M.D.: Ramanujan’s “most beautiful identity”. Am. Math. Mon. 118, 839–845 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hirschhorn, M.D., Sellers, J.A.: Elementary proofs of parity results for \(5\)-regular partitions. Bull. Aust. Math. Soc. 81, 58–63 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Mahadeva Naika, M.S., Hemanthkumar, B., Sumanth Bharadwaj, H.S.: Congruences modulo 2 for certain partition functions. Bull. Aust. Math. Soc. 93(3), 400–409 (2016)Google Scholar
  8. 8.
    Mahadeva Naika, M.S., Hemanthkumar, B., Sumanth Bharadwaj, H.S.: Color partition identities arising from Ramanujan’s theta functions. Acta Math. Vietnam. 41(4), 633–660 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Watson, G.N.: Theorems stated by Ramanujan (VII): theorems on continued fractions. J. Lond. Math. Soc. 4, 39–48 (1929)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsPES College of EngineeringMandyaIndia
  2. 2.Department of MathematicsVSK UniversityBellaryIndia

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