Let $k$ and $m$ be two coprime numbers. Let $S_{ij}=\left[ \frac{i+j}{m}\right] -\left[ \frac{i}{m}\right] -%
\left[ \frac{j}{m}\right] $, where $\left[ \text{  }\right] $ is
the Gauss symbol.( you can see Lemma4 in [Tahara](https://sci-hub.tw/https://doi.org/10.1007/BF01181625)). What is the definition of this symbol used here and what is the value of the numbers $S_{k; -k)}$ and $S_{k; k}$.

Any help would be appreciated so much. Thank you all.