Recently, partitions with fixed or bounded difference between largest and smallest parts have attracted a lot of attention. In this paper, we provide both analytic and combinatorial proofs of the generating function for k-regular partitions with bounded difference kt between largest and smallest parts. Inspired by Franklin’s result, we further find a new proof of the generating function for overpartitions with bounded part differences by using Dousse and Kim’s results on (q, z)-overGaussian polynomials.

中文翻译：

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k-具有有限的部分差异的常规分区和超分区

最近，在最大部分和最小部分之间具有固定或有界差异的分区引起了很多关注。在本文中，我们提供了k个正则分区的生成函数的解析和组合证明，其中k个正则分区的最大和最小部分之间有界差kt。受富兰克林结果的启发，我们通过使用Dousse和Kim关于（q，  z）-overGaussian多项式的结果，进一步找到了具有有限部分差异的超划分的生成函数的新证明。