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Zbl 1089.11026
Meyer, Jeffrey L.
Symmetric arguments in the Dedekind sum.
(English)
[J] Fibonacci Q. 43, No. 2, 122-123 (2005). ISSN 0015-0517

The author considers a symmetric pair $\{h,k\}$ for $h$ and $k$ relatively prime positive integers. Here a pair $\{h,k\}$ is called symmetric if $s(h, k)= s(k,h)$, where $s(h, k)$ denotes the classical Dedekind sum defined by $$s(h,k)= \sum^k_{j=1} \Biggl(\Biggl({j\over k}\Biggr)\Biggr)\Biggl(\Biggl({hj\over k}\Biggr)\Biggr)\text{ with }((x))= \cases 0\quad &\text{if }x\in\bbfZ,\\ x-[x]-{1\over 2},\quad &\text{otherwise}.\endcases$$ The main result is that $\{h, k\}$ is a symmetric pair if and only if $h= F_{2n+1}$ and $k= F_{2n+3}$ for $n\in\bbfN$ with $F_m$ the $m$th Fibonacci number.
[Kaori Ota (Tokyo)]
MSC 2000:
*11F20 Dedekind eta function, Dedekind sums
11B39 Special numbers, etc.

Keywords: Dedekind sum; symmetric pair; Fibonacci number

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