An elementary proof of the Macdonald identities for \(A_{\ell}^{(1)}\) (Q1071916)

It is shown that the Macdonald identities for \(A_{\ell}^{(1)}\) are a natural consequence of the recent multivariable generalization of classical basic hypergeometric series known as basic hypergeometric series in \(U(n)\). More precisely, a \(U(n)\) multiple series generalization of the q-binomial theorem is derived and used to generalize Cauchy's elegant proof of Jacobi's triple product identity and to give a direct, elementary proof of the Macdonald identities for \(A_{\ell}^{(1)}\).