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Given positive coprime integers $a$ and $b$ and a natural number $h$, weobtain reciprocity relations which can be used to quickly evaluate summationslike $\sum_{i=1}^{h} \{\frac{ib}{a}\}^2$ and $\sum_{i=1}^{h} \lfloor\frac{ib}{a} \rfloor^2$, where $\lfloor x \rfloor$ and $\{x\}$ denote the floorfunction and the fractional part of $x$, respectively.

Reciprocity Relations for Summations of Squares of Floor Functions and Fractional Parts of Fractions