Summation over k of Floor of x plus k over y
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Contents
Theorem
Let $x, y \in \R$ such that $y > 0$.
Then:
- $\displaystyle \sum_{0 \mathop \le k \mathop < y} \left \lfloor{x + \dfrac k y}\right \rfloor = \left \lfloor{x y + \left \lfloor{x + 1}\right \rfloor \left({\left \lceil{y}\right \rceil - y}\right)}\right \rfloor$
Proof
When $x$ increases by $1$, both sides increase by $\left \lceil{y}\right \rceil$.
So we can assume $0 \le x < 1$.
When $x = 0$, both sides are equal to $0$.
When $x$ increases past $1 - \dfrac k y$ for $0 \le k < y$, both sides increase by $1$.
Hence the result.
$\blacksquare$
Historical Note
The summation over $k$ of $\floor{x + \dfrac k y}$ is attributed to E. Busche, who published this result in $1909$.
Little biographical information can be found about him.
Sources
- 1909: E. Busche: Zur Theorie die Funktion $\left[{x}\right]$ (J. reine angew. Math. Vol. 136: 39 – 57)
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $38$