We obtain connection coefficients between q-binomial and q-trinomial coefficients. Using these, one can transform q-binomial identities into q-trinomial identities and back again. To demonstrate the usefulness of this procedure we rederive some known trinomial identities related to partition theory and prove many of the conjectures of Berkovich, McCoy and Pearce, which have recently arisen in their study of the and perturbations of minimal conformal field theory.
REFERENCES
1.
G. E. Andrews, “The theory of partitions,” Encyclopedia of Mathematics and its Applications (Addison-Wesley, Reading, MA, 1976), Vol. 2.
2.
G. E. Andrews, “q-Series: Their development and application in analysis, number theory, combinatorics, physics, and computer algebra,” in CBMS Regional Conference Series in Mathemancs (AMS, Providence, RI, 1985), Vol. 66.
3.
G. Gasper and M. Rahman, “Basic hypergeometric series,” Encyclopedia of Mathematics and its Applications (Cambridge Univiversity Press, Cambridge, 1990).
4.
P. A. MacMahon, Combinatory Analysis (Cambridge University Press, London and New York, 1916), Vol. 2.
5.
P. J.
Forrester
and G. E.
Andrews
, “Height probabilities in solid-on-solid models. I
,” J. Phys. A
19
, L923
–L926
(1986
).6.
G. E.
Andrews
and R. J.
Baxter
, “Lattice gas generalization of the hard hexagon model. III. q-Trinomial coefficients
,” J. Stat. Phys.
47
, 297
–330
(1987
).7.
G. E. Andrews, “q-Trinomial coefficients and Rogers–Ramanujan type identities,” in Analytic Number Theory, edited by B. C. Berndt, H. G. Diamond, and A. J. Hildebrand (Birkhäuser, Boston, 1990), pp. 1–11.
8.
G. E.
Andrews
, “Euler’s ‘Exemplum memorabile inductionis fallacis’ and q-trinomial coefficients
,” J. Am. Math. Soc.
3
, 653
–669
(1990
).9.
G. E.
Andrews
, “Schur’s theorem, Capparelli’s conjecture and q-trinomial coefficients
,” Contemp. Math.
166
, 141
–154
(1994
).10.
S. O.
Warnaar
and P. A.
Pearce
, “Exceptional structure of the dilute model: and Rogers–Ramanujan identities
,” J. Phys. A
27
, L891
–L897
(1994
).11.
S. O.
Warnaar
, P. A.
Pearce
, K. A.
Seaton
, and B.
Nienhuis
, “Order parameters of the dilute A models
,” J. Stat. Phys.
74
, 469
–531
(1994
).12.
G. E. Andrews, “Rogers–Ramanujan polynomials for modulus 6,” in Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam, edited by B. C. Berndt, H. G. Diamond, and A. J. Hildebrand (Birkhäuser, Boston, 1996), Vol. 1, pp. 17–30.
13.
A.
Berkovich
, B. M.
McCoy
, and W. P.
Orrick
, “Polynomial identities, indices, and duality for the superconformal model
” J. Stat. Phys.
83
, 795
–837
(1996
).14.
A.
Schilling
, “Multinomials and polynomial bosonic forms for the branching functions of the conformal coset models
,” Nucl. Phys. B
467
, 247
–271
(1996
).15.
S. O.
Warnaar
, “Fermionic solution of the Andrews–Baxter–Forrester model. II. Proof of Melzer’s polynomial identities
,” J. Stat. Phys.
84
, 49
–83
(1996
).16.
A.
Berkovich
and B. M.
McCoy
, “Generalizations of the Andrews–Bressoud identities for the superconformal model )
,” Math. Comput. Modell.
26
, 37
–49
(1997
).17.
K. A.
Seaton
and L. C.
Scott
, “q-Trinomial coefficients and the dilute A model
,” J. Phys. A
30
, 7667
–7676
(1997
).18.
S. O.
Warnaar
, “The Andrews–Gordon identities and q-multinomial coefficients
,” Commun. Math. Phys.
184
, 203
–232
(1997
).19.
G. E.
Andrews
and A.
Berkovich
, “A trinomial analogue of Bailey’s lemma and superconformal invariance
,” Commun. Math. Phys.
192
, 245
–260
(1998
).20.
A.
Berkovich
and B. M.
McCoy
, “The perturbation of the models of conformal field theory and related polynomial character identities
,” preprint ITP-SB-98-49, math.QA/9809066. To appear in the Ramanujan Journal.21.
A.
Berkovich
, B. M.
McCoy
, and P. A.
Pearce
, “The perturbations and of the minimal models and the trinomial analogue of Bailey’s lemma
,” Nucl. Phys. B
519
, 597
–625
(1998
).22.
A.
Schilling
and S. O.
Warnaar
, “Supernomial coefficients, polynomial identities and q-series
,” Ramanujan J.
2
, 459
–494
(1998
).23.
S. O.
Warnaar
, “A note on the trinomial analogue of Bailey’s lemma
,” J. Comb. Theory, Ser. A
81
, 114
–118
(1998
).24.
G. E.
Andrews
, “A polynomial identity which implies the Rogers–Ramanujan identities
,” Scr. Math.
28
, 297
–305
(1970
).25.
I. J.
Schur
, “Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche
,” S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl.
X
, 302
–321
(1917
).26.
L. J.
Slater
, “Further identities of the Rogers–Ramanujan type
,” Proc. London Math. Soc.
54
, 147
–167
(1952
).27.
H.
Göllnitz
, “Partitionen mit differenzenbedienungen
,” J. Reine Angew. Math.
225
, 154
–190
(1967
).28.
B.
Gordon
, “Some continued fractions of the Rogers–Ramanujan type
,” Duke Math. J.
31
, 741
–748
(1965
).29.
A.
Berkovich
and B. M.
McCoy
, “Continued fractions and fermionic representations for characters of minimal models
,” Lett. Math. Phys.
37
, 49
–66
(1996
).30.
A.
Berkovich
, B. M.
McCoy
, and A.
Schilling
, “Rogers–Schur–Ramanujan type identities for the minimal models of conformal field theory
,” Commun. Math. Phys.
191
, 325
–395
(1998
).31.
B. L.
Feigin
and D. B.
Fuchs
, “Verma modules over a Virasoro algebra
,” Funct. Anal. Appl.
17
, 241
–242
(1983
).32.
A. Rocha-Caridi, “Vacuum vector representation of the Virasoro algebra,” in Vertex Operators in Mathematics and Physics, edited by J. Lepowsky, S. Mandelstam, and I. M. Singer (Springer-Verlag, Berlin, 1985), pp. 451–473.
33.
V. K.
Dobrev
, “Characters of the irreducible highest weight modules over the Virasoro and super-Virasoro algebras
,” Rend. Circ. Mat. Palermo (2) Suppl.
14
, 25
–42
(1987
).34.
G. E.
Andrews
, R. J.
Baxter
, and P. J.
Forrester
, “Eight-vertex SOS model and generalized Rogers–Ramanujan-type identities
,” J. Stat. Phys.
35
, 193
–266
(1984
).35.
R. J.
Baxter
and P. J.
Forrester
, “Further exact solutions of the eight-vertex SOS model and generalizations of the Rogers–Ramanujan identities
,” J. Stat. Phys.
38
, 435
–472
(1985
).36.
G. E.
Andrews
, R. J.
Baxter
, D. M.
Bressoud
, W. H.
Burge
, P. J.
Forrester
, and G.
Viennot
, “Partitions with prescribed hook differences
,” Eur. J. Comb.
8
, 341
–350
(1987
).37.
O.
Foda
, K. S. M.
Lee
, and T. A.
Welsh
, “A Burge tree of Virasoro-type polynomial identities
,” Int. J. Mod. Phys. A
13
, 4967
–5012
(1998
).38.
A.
Berkovich
, “Fermionic counting of RSOS-states and Virasoro character formulas for the unitary minimal series Exact results
,” Nucl. Phys. B
431
, 315
–348
(1994
).39.
S. O.
Warnaar
, “Fermionic solution of the Andrews–Baxter–Forrester model. I. Unification of CTM and TBA methods
,” J. Stat. Phys.
82
, 657
–685
(1996
).40.
O.
Foda
and Y.-H.
Quano
, “Polynomial identities of the Rogers–Ramanujan type
,” Int. J. Mod. Phys. A
10
, 2291
–2315
(1995
).41.
S.
Dasmahapatra
, R.
Kedem
, T. R.
Klassen
, B. M.
McCoy
, and E.
Melzer
, “Quasi-particles, conformal field theory, and q-series
,” Int. J. Mod. Phys. B
7
, 3617
–3648
(1993
).42.
A.
Schilling
, “Polynomial fermionic forms for the branching functions of the rational coset conformal field theories
” Nucl. Phys. B
459
, 393
–436
(1996
).43.
W. N.
Bailey
, “Identities of the Rogers–Ramanujan type
,” Proc. London Math. Soc.
50
, 1
–10
(1949
).
This content is only available via PDF.
© 1999 American Institute of Physics.
1999
American Institute of Physics
You do not currently have access to this content.
