  We prove that the Fourier coefficients of a certain general eta product
considered by K. Saito are nonnegative. The proof is elementary and depends on
a multidimensional theta function identity. The z=1 case is an identity for the
generating function for p-cores due to Klyachko [17] and Garvan, Kim and
Stanton [10]. A number of other infinite products are shown to have nonnegative
coefficients. In the process a new generalization of the quintuple product
identity is derived.
