Annals of Combinatorics

, Volume 23, Issue 3–4, pp 835–888 | Cite as

Andrews–Gordon Type Series for Kanade–Russell Conjectures

  • Kağan KurşungözEmail author


We construct Andrews–Gordon type positive series as generating functions of partitions satisfying certain difference conditions in six conjectures by Kanade and Russell. Thus, we obtain q-series conjectures as companions to Kanade and Russell’s combinatorial conjectures. We construct generating functions for missing partition enumerants as well, without claiming new partition identities.


Partition generating function Andrews–Gordon identities Kanade–Russell conjectures 

Mathematics Subject Classification

05A17 05A15 11P84 



We thank George E. Andrews, Alexander Berkovich, Karl Mahlburg and Dennis Stanton for useful discussions, suggesting references or terminology during the preparation of the manuscript. The term prestidigitation and the story in the Appendix is due to the historian and my friend Emre Erol of Sabancı University. We also thank the anonymous referees for their time and their suggestions for improvements.


  1. 1.
    Alladi, K., Andrews, G.E., Gordon, B.: Refinements and generalizations of Capparelli’s conjecture on partitions. J. Algebra 174(2), 636–658 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: An analytic generalization of the Rogers-Ramanujan identities for odd moduli. Proc. Nat. Acad. Sci. U.S.A. 71, 4082–4085 (1974)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1984)CrossRefGoogle Scholar
  4. 4.
    Bringmann, K., Jennings-Shaffer, C., Mahlburg, K.: Proofs and reductions of various conjectured partition identities of Kanade and Russell. (2019)
  5. 5.
    Capparelli, S.: On some representations of twisted affine Lie algebras and combinatorial identities. J. Algebra 154(2), 335–355 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kanade, S., Russell, M.C.: IdentityFinder and some new identities of Rogers-Ramanujan type. Exp. Math. 24(4), 419–423 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    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), #P1.6 (2019)Google Scholar
  8. 8.
    Kurşungöz, K.: Parity considerations in Andrews-Gordon identities. European J. Combin. 31(3), 976–1000 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kurşungöz, K.: Cluster parity indices of partitions. Ramanujan J. 23(1-3), 195–213 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kurşungöz, K.: Andrews-Gordon type series for Capparelli’s and Göllnitz-Gordon identities. J. Combin. Theory Ser. A 165, 117–138 (2019)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Stanton, D.: The Bailey-Rogers-Ramanujan group. In: Berndt, B.C., Ono, K. (eds.) \(q\)-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000), Contemp. Math., 291, pp. 55–70. Amer. Math. Soc., Providence, RI (2001)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Engineering and Natural SciencesSabancı UniversityİstanbulTurkey

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