2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123143We 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/0607606Number TheoryCombinatorics05A30, 11F20 (Primary); 05A19, 11B65, 11F27, 11F30, 33D15 (Secondary)text