Skip to main content
SpringerLink
Log in

Polygonal numbers and Rogers–Ramanujan–Gordon theorem

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Let \(B_{k,r}(n)\) be the number of partitions of the form \(n=b_1+b_2+\cdots +b_s\), where \(b_i-b_{i+k-1}\hbox {\,\char 062\,}2\) and at most \(r-1\) of the \(b_i\) are equal to 1. In this paper, we prove that the numbers \(B_{k,r}(n)\) can be expressed in terms of Euler’s partition function p(n) considering generalized \((2m+3)\)-polygonal numbers. This result allows us to obtain new infinite families of linear partition inequalities involving p(n).

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. Andrews, G.E.: An analytic proof of the Rogers–Ramanujan–Gordon identities. Am. J. Math. 88, 844–846 (1966)

    Article  MathSciNet  Google Scholar 

  2. Andrews, G.E.: The Theory of Partitions. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  3. Andrews, G.E., Merca, M.: The truncated pentagonal number theorem. J. Comb. Theory A 119, 1639–1643 (2012)

    Article  MathSciNet  Google Scholar 

  4. Andrews, G.E., Merca, M.: Truncated theta series and a problem of Guo and Zeng. J. Comb. Theory A 154, 610–619 (2012)

    Article  MathSciNet  Google Scholar 

  5. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  6. Gordon, B.: A combinatorial generalization of the Rogers–Ramanujan identities. Am. J. Math. 83, 393–399 (1961)

    Article  MathSciNet  Google Scholar 

  7. Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. 17, 75–115 (1918)

    Article  Google Scholar 

  8. Katriel, J.: Asymptotically trivial linear homogeneous partition inequalities. J. Numb. Theory 184, 107–121 (2018)

    Article  MathSciNet  Google Scholar 

  9. Lehmer, D.H.: On the series for the partition function. Trans. Am. Math. Soc. 43, 271–295 (1938)

    Article  MathSciNet  Google Scholar 

  10. Lehmer, D.H.: On the remainders and convergence of the series for the partition function. Trans. Am. Math. Soc. 46, 362–373 (1939)

    Article  MathSciNet  Google Scholar 

  11. MacMahon, P.A.: Combinatory Analysis, vol. 2. Cambridge University Press, New York (1916)

    MATH  Google Scholar 

  12. Merca, M.: Fast algorithm for generating ascending compositions. J. Math. Model. Algorithms 11, 89–104 (2012)

    Article  MathSciNet  Google Scholar 

  13. Merca, M.: A new look on the truncated pentagonal number theorem. Carpathian J. Math. 32, 97–101 (2016)

    Article  MathSciNet  Google Scholar 

  14. Merca, M., Katriel, J.: A general method for proving the non-trivial linear homogeneous partition inequalities. Ramanujan J. (2019). https://doi.org/10.1007/s11139-018-0125-5

  15. Rademacher, H.: A convergent series for the partition function \(p(n)\). Proc. Natl. Acad. Sci. 23, 78–84 (1937)

    Article  Google Scholar 

  16. Ramanujan, S.: Proof of certain identities in combinatory analysis. Proc. Camb. Philos. Soc. 19, 214–216 (1919)

    MATH  Google Scholar 

  17. Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proc. Lond. Math. Soc. 25, 318–343 (1894)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Academy of Romanian Scientists, Ilfov 3, Sector 5, Bucharest, Romania

    Mircea Merca

Authors
  1. Mircea Merca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mircea Merca.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Merca, M. Polygonal numbers and Rogers–Ramanujan–Gordon theorem. Ramanujan J 55, 783–792 (2021). https://doi.org/10.1007/s11139-019-00242-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-019-00242-0

Keywords

Mathematics Subject Classification

Search

Navigation