K. Saito's Conjecture for Nonnegative Eta Products
  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 [12] and Garvan, Kim and Stanton [7]. 
Alexander Berkovich and Frank G. Garvan