In mathematics, the Macdonald identities are some infinite product identities associated to affine root system

Affine root system

In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebras and superalgebras, and semisimple p-adic algebraic groups, and correspond to families of Macdonald polynomials. The reduced affine root...

s, introduced by . They include as special cases the Jacobi triple product identity, Watson's quintuple product identity

Quintuple product identity

In mathematics the Watson quintuple product identity is an infinite product identity introduced by and rediscovered by and . It is analogous to the Jacobi triple product identity, and is the Macdonald identity for a certain non-reduced affine root system....

, several identities found by , and a 10-fold product identity found by .

and pointed out that the Macdonald identities are the analogs of the Weyl denominator formula for affine Kac–Moody algebra

Kac–Moody algebra

In mathematics, a Kac–Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix...

s and superalgebras.