Factorization of alternating sums of Virasoro characters

G. Andrews proved that if n is a prime number then the coefficients a k and a k + n of the product ( q , q ) ∞ / ( q n , q n ) ∞ = ∑ k a k q k have the same sign, see [G. Andrews, On a conjecture of Peter Borwein, J. Symbolic Comput. 20 (1995) 487–501]. We generalize this result in several directions. Our results are based on the observation that many products can be written as alternating sums of characters of Virasoro modules.