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In this paper, we define multiple Dedekind sums by products of Bernoulli functions. From the Fourie expansions of Bernoulli functions, we express the Dedekind sums as series representatios. Then by a combinatorial-geometric method, we give a new proof of a Knopp-type identity for the Dedekind sums. p ≥ ∈ Z , let () p BX be the p th Bernoulli polynomial and

An Analytic and Combinatorial-Geometric Proof of a Knopp-Type Identity for Multiple Dedekind Sums