Macdonald identities

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In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by Ian Macdonald (1972). They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several identities found by Dyson (1972), and a 10-fold product identity found by Winquist (1969).

Kac (1974) and Moody (1975) pointed out that the Macdonald identities are the analogs of the Weyl denominator formula for affine Kac–Moody algebras and superalgebras.


  • Demazure, Michel (1977), "Identités de Macdonald", Séminaire Bourbaki, 28e année (1975/1976), Exp. No. 483, Lecture Notes in Math, 567, Berlin, New York: Springer-Verlag, pp. 191–201, MR 0476815
  • Dyson, Freeman J. (1972), "Missed opportunities", Bulletin of the American Mathematical Society, 78: 635–652, doi:10.1090/S0002-9904-1972-12971-9, ISSN 0002-9904, MR 0522147
  • Kac, Victor G (1974), "Infinite-dimensional Lie algebras, and the Dedekind η-function", Akademija Nauk SSSR. Funkcionalnyi Analiz i ego Priloženija, 8 (1): 77–78, doi:10.1007/BF02028313, ISSN 0374-1990, MR 0374210
  • Moody, R. V. (1975), "Macdonald identities and Euclidean Lie algebras", Proceedings of the American Mathematical Society, 48: 43–52, doi:10.2307/2040690, ISSN 0002-9939, JSTOR 2040690, MR 0442048
  • Macdonald, I. G. (1972), "Affine root systems and Dedekind's η-function", Inventiones Mathematicae, 15: 91–143, doi:10.1007/BF01418931, ISSN 0020-9910, MR 0357528
  • Winquist, Lasse (1969), "An elementary proof of p(11m+6) ≡ 0 mod 11", Journal of Combinatorial Theory, 6: 56–59, doi:10.1016/s0021-9800(69)80105-5, MR 0236136