Hermite's identity

In mathematics, the Hermite's identity states that for every real number "x" and positive integer "n" the following holds::$sum\_\{k=0\}^\{n-1\}leftlfloor\; x+frac\{k\}\{n\}\; ight\; floor=lfloor\; nx\; floor\; .$

Proof

Write $x=lfloor\; x\; floor+\{x\}$. There is exactly one $k\text{'}in\{1,...,n\}$ with

Now $sum\_\{k=0\}^\{n-1\}leftlfloor\; x+frac\{k\}\{n\}\; ight\; floor=sum\_\{k=0\}^\{k\text{'}-1\}\; lfloor\; x\; floor+sum\_\{k=k\text{'}\}^\{n-1\}\; (lfloor\; x\; floor+1)=n,\; lfloor\; x\; floor+n-k\text{'}=n,\; lfloor\; x\; floor+lfloor\; n,\{x\}\; floor=leftlfloor\; n,\; [x]\; +n,\; \{x\}\; ight\; floor=lfloor\; nx\; floor$

See also

Floor function