Any hints how to compute this sum $$\sum_{i=1}^{N1}\left[i\frac{K}{N}\right]^{p}?$$ where K < N , $\left[\cdot\right]$ denotes fractional part, $p\in N$

The article On Certain Sums of Fractional Parts by Gandhi and Williams answers your question for $p=1$; it's likely that since 1974 this result has been generalized, but I wasn't able to find a reference. 

