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, 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.

中文翻译：

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素数的模数幂数的模幂

集成的功率系列被称为缺位模中号，如果几乎所有的系数是整除中号。受分区函数奇偶性问题的启发，Gordon和Ono研究了t型规则分区的生成函数，并确定了这些函数何时是素数的模幂的幂。我们通过研究称为Dedekindη商和广义Dedekindη商的无穷乘积，以多种方式推广他们的结果。然后，我们将结果应用于Nekrasov，Okounkov和Han考虑的分区函数的生成函数。