Skip to main content
SpringerLink
Log in

On identities of the Rogers-Ramanujan type

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

A generalized Bailey pair, which contains several special cases considered by Bailey (Proc. London Math. Soc. (2), 50, 421–435 (1949)), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated q-difference equations points to a connection with a mild extension of Gordon’s combinatorial generalization of the Rogers-Ramanujan identities (Amer. J. Math., 83, 393–399 (1961)). This, in turn, allows the formulation of natural combinatorial interpretations of many of the identities in Slater’s list (Proc. London Math. Soc. (2) 54, 147–167 (1952)), as well as the new identities presented here. A list of 26 new double sum–product Rogers-Ramanujan type identities are included as an Appendix.

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.: A generalization of the Göllnitz-Gordon partition theorems. Proc. Amer. Math. Soc. 18(5), 945–952 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andrews, G.E.: The Theory of Partitions. Addison-Wesley (1976); reissued Cambridge Univ. Press (1998)

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

    Article  MATH  Google Scholar 

  4. Andrews, G. E.: Problems and Prospects for basic hypergeometric functions. In Askey, R. (ed.) Theory and Application of Special Functions, pp. 191–214, Academic Press, New York (1975)

    Google Scholar 

  5. Andrews, G. E.: Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114, 267–283 (1984)

    MATH  MathSciNet  Google Scholar 

  6. Andrews, G. E.: q-series: Their development and application in analysis, number theory, combinatorics, physics, and computer algebra, CBMS Regional Conferences Series in Mathematics, no. 66, American Mathematical Society, Providence, RI (1986)

    Google Scholar 

  7. Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight vertex SOS model and generalized Rogers-Ramanujan type identities. J. Statist. Phys. 35, 193–266 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bailey, W.N.: Some identities in combinatory analysis. Proc. London Math. Soc. (2) 49, 421–435 (1947)

    MATH  MathSciNet  Google Scholar 

  9. Bailey, W.N.: Identities of the Rogers-Ramanujan type. Proc. London Math. Soc. (2) 50, 1–10 (1949)

    MathSciNet  Google Scholar 

  10. Berkovich, A., McCoy, B.M.: Continued fractions and fermionic representations for characters of M(p,p’) minimal models. Lett. Math. Phys. 37, 49–66 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Berkovich, A., McCoy, B.M., Orrick, W.P.: Polynomial identities, indices, and duality for the N=1 superconformal model SM(2,4). J. Statist. Phys. 83, 795–837 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Berkovich, A., McCoy, B.M., Schilling, A.: Rogers-Schur-Ramanujan type identities for the M(p,p’) minimal models of conformal field theory. Comm. Math. Phys. 191, 211–223 (1998)

    Article  MathSciNet  Google Scholar 

  13. Corteel, S., Lovejoy, J.: Overpartitions. Trans. Amer. Math Soc. 356, 1623–1635 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gasper, G., Rahman, M.: Basic hypergeometric series. Cambridge Univ. Press (1990)

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

    MATH  MathSciNet  Google Scholar 

  16. Lovejoy, J.: Gordon’s theorem for overpartitions. J. Comb. Theory Ser. A 103, 393–401 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. MacMahon, P.A.: Combinatory analysis. 2, Cambridge Univ. Press, London (1918)

    Google Scholar 

  18. Paule, P.: Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type. Electron. J. Combin. 1, # R10, 1–9 (1994)

    MATH  MathSciNet  Google Scholar 

  19. Petkovsek, M., Wilf, H.S., Zeilberger, D.: A =B. A. K. Peters, Wellesley, MA (1996)

  20. Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proc. London Math Soc. (1) 25, 318–343 (1894)

    MATH  Google Scholar 

  21. Rogers, L.J.: On two theorems of combinatory analysis and some allied identities. Proc. London Math. Soc. (2) 16, 315–336 (1917)

    MATH  Google Scholar 

  22. Schilling, A., Warnaar, S.O.: Supernomial coefficients, polynomial identities, and q-series. Ramanujan J. 2, 459–494 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Schur, I.: Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche. Sitzungsberichte der Berliner Akademie, 302–321 (1917)

  24. Sills, A.V.: Finite Rogers Ramanujan type identities. Electronic J. Combin. 10(1), # R13, 1–122 (2003)

    MathSciNet  Google Scholar 

  25. Sills, A.V.: RRtools—a Maple package for the discovery and proof of Rogers-Ramanujan type identities. J. Symbolic Comput. 37, 415–448 (2004)

    Article  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  27. Warnaar, S.O.: The generalized Borwein conjecture II: refined q-trinomial coefficients. Discrete Math. 272, 215–258 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Warnaar, S.O.: q-trinomial identities. J. Math. Phys. 40(4), 2514–2530 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  29. Warnaar, S.O.: Refined q-trinomial coefficients and character identities. J. Stat. Phys. 102, 1065–1081 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. Wilf, H.S., Zeilberger, D.: Rational function certification of hypergeometric multi- integral/sum/qidentities. Invent. Math. 108, 575–633 (1992)

    Article  MathSciNet  Google Scholar 

  31. Zeilberger, D.: A fast algorithm for proving termininating hypergeometric identities. Discrete Math. 80, 207–211 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Rutgers University, Hill Center–Busch Campus, Piscataway, New Jersey, 08854

    Andrew V. Sills

Authors
  1. Andrew V. Sills
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrew V. Sills.

Additional information

2000 Mathematics Subject Classification Primary—11B65; Secondary—11P81, 05A19, 39A13

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sills, A.V. On identities of the Rogers-Ramanujan type. Ramanujan J 11, 403–429 (2006). https://doi.org/10.1007/s11139-006-8483-9

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-006-8483-9

Keywords

Search

Navigation