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On a reciprocity law for finite multiple zeta values. (English) Zbl 1232.11088
Summary: It was shown by Kirschenhofer and Prodinger [Comb. Probab. Comput. 7, No. 1, 111–120 (1998; Zbl 0892.68021)] and Kuba et al. [Integers 8, No. 1, Article A17, 20 p., electronic only (2008; Zbl 1202.68492)] that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that this reciprocity relation can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a combinatorial proof of the shuffle identity based on partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.
11M32 Multiple Dirichlet series and zeta functions and multizeta values
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