K. Saito's Conjecture for Nonnegative Eta Products and Analogous Results for Other Infinite Products
arXiv:math/0702027
Abstract
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.
15 pages; greatly expanded version of the earlier 8 page paper math.NT/0607606