Skip to main content
SpringerLink
Log in

New proofs of some double sum Rogers–Ramanujan type identities

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Recently, Rosengren utilized an integral method to prove a number of conjectural identities found by Kanade and Russell. Using this integral method, we give new proofs to some double sum identities of Rogers–Ramanujan type. These identities were earlier proved by approaches such as combinatorial arguments or using q-difference equations. Our proofs are based on streamlined calculations, which relate these double sum identities to some known Rogers–Ramanujan type identities with single sums. Moreover, we prove a conjectural identity of Andrews and Uncu which was earlier confirmed by Chern.

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.: On the General Rogers–Ramanujan Theorem. American Mathematical Society, Providence (1974)

    Book  MATH  Google Scholar 

  2. Andrews, G.E.: The Theory of Partitions. Addison-Wesley, Reading (1976). Reissued Cambridge (1998)

  3. Andrews, G.E.: Schur’s theorem, Capparelli’s conjecture and \(q\)-trinomial coefficients. In: Andrews, G.E., Bressoud, D.M., Parson, L.A. (eds.) The Rademacher Legacy to Mathematics, pp. 141–154. American Mathematical Society, Providence (1994)

    Chapter  Google Scholar 

  4. Andrews, G.E.: Sequences in partitions, double \(q\)-series and the mock theta function \(\rho _3(q)\). In: Pillwein, V., Schneider, C. (eds.) Algorithmic Combinatorics—Enumerative Combinatorics, Special Functions and Computer Algebra, pp. 25–46. Springer, Cham (2020)

    Google Scholar 

  5. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, Part II. Springer, New York (2009)

  6. Andrews, G.E., Bringmann, K., Mahlburg, K.: Double series representations for Schur’s partition function and related identities. J. Combin. Theory Ser. A 132, 102–119 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Andrews, G.E., Uncu, A.K.: Sequences in overpartitions. arXiv:2111.15003v1

  8. Bressoud, D.M.: A generalization of the Rogers–Ramanujan identities for all moduli. J. Combin. Theory Ser. A 27, 64–68 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bringmann, K., Jennings-Shaffer, C., Mahlburg, K.: Proofs and reductions of various conjectured partition identities of Kanade and Russell. J. Reine Angew. Math. 766, 109–135 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cao, Z., Wang, L.: Multi-sum Rogers–Ramanujan type identities, arXiv:2205.12786

  11. Capparelli, S.: On some representations of twisted affine Lie algebras and combinatorial identities. J. Algebra 154, 335–355 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chan, H.C.: Ramanujan’s cubic continued fraction and a generalization of his “most beautiful identity.” Int. J. Number Theory 6, 673–680 (2010)

  13. Chern, S.: Private communication on March 29, 2022

  14. Chern, S.: Asymmetric Rogers–Ramanujan type identities. I. The Andrews–Uncu Conjecture, submitted, arXiv:2203.15168

  15. Gasper, G., Rahman, M.: Encyclopedia of Mathematics and Its Applications, vol. 96. Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004)

  16. Göllnitz, H.: Partitionen mit Differenzenbedingungen. J. Reine Angew. Math. 225, 154–190 (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. Hirschhorn, M.D., Garvan, F., Borwein, J.: Cubic analogs of the Jacobian cubic theta function \(\theta (z, q)\). Can. J. Math. 45, 673–694 (1993)

    Article  MATH  Google Scholar 

  19. Kanade, S., Russell, M.C.: IdentityFinder and some new partition identities of Rogers–Ramanujan type. Exp. Math. 24(4), 419–423 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kanade, S., Russell, M.C.: Staircases to analytic sum-sides for many new integer partition identities of Rogers–Ramanujan type. Electron. J. Combin. 26, 1–6 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kurşungöz, K.: Andrews-Gordon type series for Capparelli’s and Göllnitz-Gordon identities. J. Combin. Theory Ser. A 165, 117–138 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. McLaughlin, J.: Some more identities of Kanade–Russell type derived using Rosengren’s method. Ann. Combin. (2022). https://doi.org/10.1007/s00026-022-00586-3

  23. McLaughlin, J., Sills, A.V., Zimmer, P.: Rogers–Ramanujan–Slater type identities. Electron. J. Combin. 15, #DS15 (2008)

  24. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)

    MATH  Google Scholar 

  25. Rahman, M.: Some cubic summation formulas for basic hypergeometric series. Utilitas Math. 36, 161–172 (1989)

    MathSciNet  MATH  Google Scholar 

  26. Rosengren, H.: Proofs of some partition identities conjectured by Kanade and Russell. Ramanujan J. https://doi.org/10.1007/s11139-021-00389-9

  27. Sills, A.V.: An Invitation to the Rogers–Ramanujan Identities. CRC Press, Boca Raton (2018)

  28. Slater, L.J.: Further identities of the Rogers–Ramanujan type. Proc. Lond. Math. Soc. (2) 54(1), 147–167 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  29. Takigiku, M., Tsuchioka, S.: A proof of conjectured partition identities of Nandi (2019). arXiv:1910.12461

  30. Uncu, A., Zudilin, W.: Reflecting (on) the modulo 9 Kanade–Russell (conjectural) identities. arXiv:2106.02959v2

  31. Xia, E.X.W., Yao, O.X.M.: Analogues of Ramanujan’s partition identities. Ramanujan J. 31, 373–396 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Matthew Russell for pointing out the relations between (4.13)–(4.14) and the conjectural identities \(I_5\) and \(I_6\) in [19] and their discussions of these identities in [20, Sects. 5.1.1 and 5.1.2]. We also thank the referee for his/her helpful comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, People’s Republic of China

    Liuquan Wang

Authors
  1. Liuquan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liuquan Wang.

Additional information

Publisher's Note

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

This work was supported by the National Natural Science Foundation of China (Grant No. 12171375).

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, L. New proofs of some double sum Rogers–Ramanujan type identities. Ramanujan J 62, 251–272 (2023). https://doi.org/10.1007/s11139-022-00654-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-022-00654-5

Keywords

Mathematics Subject Classification

Search

Navigation