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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Applications 2, Addison-Wesley, Reading, MA, 1976.
G. E. Andrews, Schur’s theorem, Capparelli’s conjecture and q-trinomial coefficients, Contemp. Math., 166 (1994), 141–154.
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.
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.
W. N. Bailey, Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2), 50 (1949), 1–10.
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.
W. H. Burge, Restricted partition pairs, J. Combin. Theory Ser. A, 63 (1993), 210–222.
L. M. Butler, Subgroup lattices and symmetric functions, Mem. Amer. Math. Soc., 112 (1994), no. 539.
L. Carlitz, Remark on a combinatorial identity, J. Combin. Theory Ser. A, 17 (1974), 256–257.
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.
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.
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.
G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications 35, Cambridge University Press Cambridge, 1990.
I. M. Gessel and C. Krattenthaler, Cylindric partitions, Trans. Amen Math. Soc., 349 (1997), 429–479.
H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math., 225 (1967), 154–190.
B. Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J., 31 (1965), 741–748.
H. W. Gould, A new symmetrical combinatorial identity, J. Combin. Theory Ser. A, 13 (1972), 278–286.
A. N. Kirillov, Dilogarithm identities, Prog. Theoret. Phys. Suppl., 118 (1995), 61–142.
J. Lepowsky and M. Prime, Structure of the standard modules for the affine Lie algebra A 1 (1) Contemporary Mathematics 46, AMS, Providence, 1985.
A. Nakayashiki and Y. Yamada, Crystallizing the spinon basis, Comm. Math. Phys., 178 (1996), 179–200.
A. Nakayashiki and Y. Yamada, Crystalline spinon basis for RSOS models, Internat. J. Modern Phys. A, 11 (1996), 395–408.
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.
A. Schilling and S. O. Waraaar, A higher-level Bailey lemma, Internat. J. Modern Phys. B, 11 (1997), 189–195.
A. Schilling and S. O. Warnaar, A higher level Bailey lemma: Proof and application, Ramanujan J., 2 (1998), 327–349.
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.
L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2), 54 (1952), 147–167.
S. O. Warnaar, The Andrews-Gordon identities and q-multinomial coefficients, Comm. Math. Phys., 184 (1997), 203–232.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media New York
About this chapter
Cite this chapter
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
Download citation
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