Generalized mth order Jacobi theta functions and the Macdonald identities

Abstract

We describe an m th order generalization of Jacobi's theta functions and use these functions to construct classes of theta function identities in multiple variables. These identities are equivalent to the Macdonald identities for the seven infinite families of irreducible affine root systems. They are also equivalent to some elliptic determinant evaluations proven recently by Rosengren and Schlosser. © 2008 World Scientific Publishing Company.