  An integral power series is called lacunary modulo $M$ if almost all of its
coefficients are divisible by $M$. Motivated by the parity problem for the
partition function, $p(n)$, Gordon and Ono studied the generating functions for
$t$-regular partitions, and determined conditions for when these functions are
lacunary modulo powers of primes. We generalize their results in a number of
ways by studying infinite products called Dedekind eta-quotients and
generalized Dedekind eta-quotients. We then apply our results to the generating
functions for the partition functions considered by Nekrasov, Okounkov, and
Han.
