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

Abstract

Given positive coprime integers $a$ and $b$ and a natural number $h$, we obtain reciprocity relations which can be used to quickly evaluate summations like ${\sum }_{i=1}^h\{\frac{ib}{a}{\}}^2$ and ${\sum }_{i=1}^h\lfloor \frac{ib}{a}{\rfloor }^2$, where $\lfloor x\rfloor$ and $\{x\}$ denote the floor function and the fractional part of $x$, respectively.