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A Generalization of the q-Saalschütz Sum and the Burge Transform

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Physical Combinatorics

Part of the book series: Progress in Mathematics ((PM,volume 191))

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Abstract

A generalization of the q-(Pfaff-)Saalschütz summation formula is proved. This implies a generalization of the Burge transform, resulting in an additional dimension of the “Burge tree”. Limiting cases of our summation formula imply the (higher-level) Bailey lemma, provide a new decomposition of the q-multinomial coefficients, and can be used to prove the Lepowsky and Primc formula for the A (1)1 string functions.

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References

  1. G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Applications 2, Addison-Wesley, Reading, MA, 1976.

    Google Scholar 

  2. G. E. Andrews, Schur’s theorem, Capparelli’s conjecture and q-trinomial coefficients, Contemp. Math., 166 (1994), 141–154.

    Article  Google Scholar 

  3. G. E. Andrews and R. J. Baxter, Lattice gas generalization of the hard hexagon model III: q-Trinomial coefficients, J. Statist. Phys., 47 (1987), 297–330.

    Article  MathSciNet  MATH  Google Scholar 

  4. T. Arakawa, T. Nakanishi, K. Oshima, and A. Tsuchiya, Spectral decomposition of path space in solvable lattice model, Comm. Math. Phys., 181 (1996), 157–182.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  6. P. Bouwknegt, A. W. W. Ludwig, and K. Schoutens, Spinon basis for higher level SU (2) WZW models, Phys. Lett. B, 359 (1995), 304–312.

    Article  MathSciNet  Google Scholar 

  7. W. H. Burge, Restricted partition pairs, J. Combin. Theory Ser. A, 63 (1993), 210–222.

    Article  MathSciNet  MATH  Google Scholar 

  8. L. M. Butler, Subgroup lattices and symmetric functions, Mem. Amer. Math. Soc., 112 (1994), no. 539.

    Google Scholar 

  9. L. Carlitz, Remark on a combinatorial identity, J. Combin. Theory Ser. A, 17 (1974), 256–257.

    Article  MATH  Google Scholar 

  10. E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, Exactly solvable SOS models: Local height probabilities and theta function identities, Nuclear Phys. B, 290 (1987), 231–273.

    Article  MathSciNet  MATH  Google Scholar 

  11. E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, Exactly solvable SOS models II: Proof of the star-triangle relation and combinatorial identities, Adv. Stud. Pure Math., 16 (1988), 17–122.

    MathSciNet  Google Scholar 

  12. O. Foda, K. S. M. Lee, and T. A. Welsh, A Burge tree of Virasoro-type polynomial identities, Internat. J. Modern Phys. A, 13 (1998), 4967–5012.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications 35, Cambridge University Press Cambridge, 1990.

    Google Scholar 

  14. I. M. Gessel and C. Krattenthaler, Cylindric partitions, Trans. Amen Math. Soc., 349 (1997), 429–479.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math., 225 (1967), 154–190.

    MathSciNet  MATH  Google Scholar 

  16. B. Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J., 31 (1965), 741–748.

    Article  Google Scholar 

  17. H. W. Gould, A new symmetrical combinatorial identity, J. Combin. Theory Ser. A, 13 (1972), 278–286.

    Article  MATH  Google Scholar 

  18. A. N. Kirillov, Dilogarithm identities, Prog. Theoret. Phys. Suppl., 118 (1995), 61–142.

    Article  MathSciNet  Google Scholar 

  19. J. Lepowsky and M. Prime, Structure of the standard modules for the affine Lie algebra A 1 (1) Contemporary Mathematics 46, AMS, Providence, 1985.

    Google Scholar 

  20. A. Nakayashiki and Y. Yamada, Crystallizing the spinon basis, Comm. Math. Phys., 178 (1996), 179–200.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Nakayashiki and Y. Yamada, Crystalline spinon basis for RSOS models, Internat. J. Modern Phys. A, 11 (1996), 395–408.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Schilling, Multinomials and polynomial bosonic forms for the branching functions of the >183-01 conformal coset models, Nuclear Phys. B, 467 (1996), 247–271.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Schilling and S. O. Waraaar, A higher-level Bailey lemma, Internat. J. Modern Phys. B, 11 (1997), 189–195.

    Article  MATH  Google Scholar 

  24. A. Schilling and S. O. Warnaar, A higher level Bailey lemma: Proof and application, Ramanujan J., 2 (1998), 327–349.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Schilling and S. O. Warnaar, Conjugate Bailey pairs: From configuration sums and fractional-level string functions to Bailey’s lemma, preprint math.QA/9906092.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. S. O. Warnaar, The Andrews-Gordon identities and q-multinomial coefficients, Comm. Math. Phys., 184 (1997), 203–232.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Instituut voor Theoretische Fysica, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE, Amsterdam, Netherlands

    Anne Schilling & S. Ole Warnaar

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Anne Schilling

Authors
  1. Anne Schilling
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  2. S. Ole Warnaar
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Editor information

Editors and Affiliations

  1. Research Institute for Mathematical Sciences (RIMS), Kyoto University, 606-01, Kyoto, Japan

    Masaki Kashiwara  & Tetsuji Miwa  & 

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Schilling, A., Warnaar, S.O. (2000). A Generalization of the q-Saalschütz Sum and the Burge Transform. In: Kashiwara, M., Miwa, T. (eds) Physical Combinatorics. Progress in Mathematics, vol 191. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1378-9_5

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  • DOI: https://doi.org/10.1007/978-1-4612-1378-9_5

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-7121-5

  • Online ISBN: 978-1-4612-1378-9

  • eBook Packages: Springer Book Archive

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