Generalization of the non-local derangement identity and applications to multiple zeta-type series

Abstract

The goal of this paper is the study of a transformation concerning the general K-fold finite sums of the form

$$\begin{aligned} \sum _{N\ge n_1\ge \cdots \ge n_K\ge 1}\frac{1}{b_{n_K}}\cdot \prod _{j=1}^{K-1}\frac{1}{a_{n_j}}, \end{aligned}$$

where $(K,N)\in \mathbb {N}^2$ and $\{a_n\}_{n=1}^{\infty }$, $\{b_n\}_{n=1}^{\infty }$ are appropriate real sequences. In the application part of our paper we apply the developed transformation to two special parametric multiple zeta-type series that generalize the well-know formula $\zeta ^\star (\{2\}_K,1)=2\zeta (2K+1)$, $K\in \mathbb {N}$. As a corollary of our parametric results, we also prove several sum formulas involving multiple zeta-star values.