techniques. They proved these identities using level 3 modules for the affine Lie algebra . In the course of this proof they invented and used what they called -algebras. Lepowsky and Wilson's approach is universal, in that it is able to treat all affine Lie algebras at all levels. It can be used to find (and prove) new partition identities. First such example is that of Capparelli's identities discovered by Stefano Capparelli using level 3 modules for the affine Lie algebra . See also Rogers polynomials Continuous q-Hermite polynomials References W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge. George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. . Bruce C. Berndt, Heng Huat Chan, Sen-Shan Huang, Soon-Yi Kang, Jaebum Sohn, Seung Hwan Son, The Rogers-Ramanujan Continued Fraction, J. Comput. Appl. Math. 105 (1999), pp. 9–24. Cilanne Boulet, Igor Pak, A Combinatorial Proof of the Rogers-Ramanujan and Schur Identities, Journal of Combinatorial Theory, Ser. A, vol. 113 (2006), 1019–1030. James Lepowsky and Robert L. Wilson, Construction of the affine Lie algebra , Comm. Math. Phys. 62 (1978) 43-53. James Lepowsky and Robert L. Wilson, A new family of algebras underlying the Rogers-Ramanujan identities, Proc. Natl. Acad. Sci. USA 78 (1981), 7254-7258. James Lepowsky and Robert L. Wilson, The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities, Invent. Math. 77 (1984), 199-290. James Lepowsky and Robert L. Wilson, The structure of standard modules, II: The case , principal gradation, Invent. Math. 79 (1985), 417-442. Stefano Capparelli, Vertex operator relations for affine algebras and combinatorial identities'', Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick. 1988. 107 pp. External links Hypergeometric functions Integer partitions Mathematical identities Q-analogs Modular forms Srinivasa Ramanujan