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.
References Demazure, Michel (1977), "Identités de Macdonald", Séminaire Bourbaki, 28e année (1975/1976), Exp. No. 483, Lecture Notes in Mathematics, vol. 567, Berlin, New York: Springer-Verlag, pp. 191–201, doi:10.1007/BFb0096070, ISBN 978-3-540-08064-0, MR 0476815 Dyson, Freeman J. (1972), "Missed opportunities", Bulletin of the American Mathematical Society, 78 (5): 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", Funktsional'nyi Analiz i Ego Prilozheniya, 8 (1): 77–78, MR 0374210 Translated in Kats, Victor G. (1974), "Infinite-dimensioned Lie algebras and Dedekind'sη-function", Functional Analysis and Its Applications, 8: 68–70, doi:10.1007/BF02028313 Moody, R. V. (1975), "Macdonald identities and Euclidean Lie algebras", Proceedings of the American Mathematical Society, 48 (1): 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 (2): 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