We focus on writing double sum representations of the generating functions for the number of partitions satisfying some gap conditions. Some example sets of partitions to be considered are partitions into distinct parts and partitions that satisfy the gap conditions of the Rogers–Ramanujan, Göllnitz–Gordon, and little Göllnitz theorems. We refine our representations by imposing a bound on the largest part and find finite analogues of these new representations. These refinements lead to many q-series and polynomial identities. Additionally, we present a different construction and a double sum representation for the products similar to the ones that appear in the Rogers–Ramanujan identities.

我们专注于编写满足某些间隙条件的分区数量的生成函数的双和表示。要考虑的一些示例分区集是划分为不同部分和满足 Rogers-Ramanujan、Göllnitz-Gordon 和 little Göllnitz 定理的间隙条件的分区。我们通过对最大部分施加界限来改进我们的表示，并找到这些新表示的有限类似物。这些改进导致了许多q级数和多项式恒等式。此外，我们为与 Rogers-Ramanujan 恒等式中出现的产品类似的产品提供了不同的构造和双和表示。