Dedekind type DC sums and their generalizations are defined in terms of Euler functions and their generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind type DC sums by replacing the Euler function appearing in Dedekind sums, and they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds of new generalizations of the poly-Dedekind type DC sums. One is a unipoly-Dedekind type DC sum associated with the type 2 unipoly-Euler functions expressed in the type 2 unipoly-Euler polynomials using the modified polyexponential function, and we study some identities and the reciprocity relation for these unipoly-Dedekind type DC sums. The other is a unipoly-Dedekind sums type DC associated with the poly-Euler functions expressed in the unipoly-Euler polynomials using the polylogarithm function, and we derive some identities and the reciprocity relation for those unipoly-Dedekind type DC sums.

中文翻译：

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单多边形-戴德金型DC和的一些恒等式和互易关系

Dedekind 型 DC 和及其泛化是根据 Euler 函数及其泛化定义的。最近，马等人。(Adv. Differ. Equ. 2021:30 2021) 通过替换出现在 Dedekind 和中的 Euler 函数引入了 poly-Dedekind 型 DC 和，并且证明它们满足互易关系。在本文中，我们考虑了 poly-Dedekind 型 DC 和的两种新的推广。一个是与使用修正的多指数函数在类型 2 unipoly-Euler 多项式中表示的类型 2 unipoly-Euler 函数相关联的 unipoly-Dedekind 型 DC 和，我们研究了这些 un​​ipoly-Dedekind 型 DC 和的一些恒等式和互易关系.