Skip to main content
SpringerLink
Log in

Basic series identities and combinatorics

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Analogous to P.A. MacMahon’s combinatorial interpretations of the Rogers–Ramanujan identities, we interpret two basic series identities combinatorially in two different ways—using split \((n+t)\)-color partitions and the modified lattice paths. This leads to two new 3-way combinatorial identities. We conclude by posing three significant open problems.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Agarwal, A.K.: Partitions with “N copies Of N”. Lecture Notes in Mathematics, vol. 1234. Springer, New York (1985)

    MATH  Google Scholar 

  2. Agarwal, A.K.: A note on \(n(x, y)\)-reflected lattice paths. Fibonacci Q. 25(4), 317–319 (1987)

    MathSciNet  MATH  Google Scholar 

  3. Agarwal, A.K.: Rogers–Ramanujan identities for \(n\)-color partitions. J. Number Theory 28, 299–305 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Agarwal, A.K.: New combinatorial interpretations of two analytic identities. Proc. Am. Math. Soc. 107(2), 561–567 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Agarwal, A.K.: New classes of infinite 3-way partition identities. ARS Comb. 44, 33–54 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Agarwal, A.K., Andrews, G.E.: Hook differences and lattice paths. J. Stat. Plan. Inference 14(1), 5–14 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Agarwal, A.K., Andrews, G.E.: Rogers–Ramanujan identities for partitions with “N copies of N”. J. Comb. Theory Ser. A 45, 40–49 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Agarwal, A.K., Bressoud, D.M.: Lattice paths and multiple basic hypergeometric series. Pac. J. Math. 136(2), 209–228 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Agarwal, A.K., Rana, M.: New combinatorial versions of \({\text{ G }}\ddot{\text{ o }}{\text{ llnitz }}\)–Gordon identities. Util. Math. 79, 145–155 (2009)

    MathSciNet  Google Scholar 

  10. Agarwal, A.K., Sood, G.: Split \((n+t)\) color partitions and Gordon–McIntosh eight order mock theta functions. Electron J. Comb. 21(2), P2-46 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Agarwal, A.K., Andrews, G.E., Bressoud, D.M.: The Bailey lattice. J. Indian Math. Soc. 51, 57–73 (1987)

    MathSciNet  MATH  Google Scholar 

  12. Andrews, G.E.: An analytic generalization of the Rogers–Ramanujan identities for odd moduli. Proc. Nat. Acad. Sci. USA 71, 4082–4085 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chu, W., Zhang, W.: Bilateral Bailey lemma and Rogers–Ramanujan identities. Adv. Appl. Math. 42, 358–391 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Connor, W.G.: Partition theorems related to some identities of Rogers and Watson. Trans. Am. Math. Soc. 214, 95–111 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  15. Göllnitz, H: Einfache Partitionen (Unpublished), Diplomarbeit W.s., Gottingen (1960)

  16. Göllnitz, H.: Partitionen mit Differenzenben-dingungen. J. Reine Angew. Math. 225, 154–90 (1967)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Gordon, B.: Some continued fractions of the Rogers–Ramanujan type. Duke J. Math. 32, 741–748 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gordon, B., McIntosh, R.J.: Some eight order mock theta functions. J. Lond. Math. Soc. 62(2), 321–335 (2000)

    Article  MATH  Google Scholar 

  20. Goyal, M., Agarwal, A.K.: Further Rogers–Ramanujan identities for \(n\)-color partitions. Util. Math. 95, 141–148 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Hirschhorn, M.D.: Some partition theorems of the Rogers–Ramanujan type. J. Comb. Theory Ser. A 27(1), 33–37 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. MacMahon, P.A.: Combinatory Analysis, vol. 2. Cambridge University Press, London (1916)

    MATH  Google Scholar 

  23. Polya, G.: On the number of certain lattice polygons. J. Comb. Theory 6, 102–105 (1969)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  25. Sachdeva, R., Agarwal, A.K.: Modified lattice paths and Gordon–McIntosh eight order mock theta functions. Communicated

  26. Slater, L.J.: Further identities of the Rogers–Ramanujan type. Proc. Lond. Math. Soc. 54(2), 147–167 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  27. Subbarao, M.V.: Some Rogers–Ramanujan type partition theorems. Pac. J. Math. 120, 431–435 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  28. Subbarao, M.V., Agarwal, A.K.: Further theorems of Rogers–Ramanujan type. Can. Math. Bull. 31(2), 210–214 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, 160014, India

    A. K. Agarwal & R. Sachdeva

Authors
  1. A. K. Agarwal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Sachdeva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. K. Agarwal.

Additional information

A.K. Agarwal—Supported by UGC Emeritus Fellowship No. Emeritus-2014-15-GEN-4075/(SA-II). R. Sachdeva—Supported by a Senior Research Fellowship, UGC Research Grant No. F.17-7/J/2004(SA-I).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Agarwal, A.K., Sachdeva, R. Basic series identities and combinatorics. Ramanujan J 42, 725–746 (2017). https://doi.org/10.1007/s11139-015-9754-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-015-9754-0

Keywords

Mathematics Subject Classification

Search

Navigation