Hermite's identity

In mathematics, Hermite's identity, named after Charles Hermite, gives the value of a summation involving the floor function. It states that for every real number x and for every positive integer n the following identity holds:^[1]^[2]

\sum_{k=0}^{n-1}\left\lfloor x+\frac{k}{n}\right\rfloor=\lfloor nx\rfloor .

Proof

Split $x$ into its integer part and fractional part, $x=\lfloor x\rfloor+\{x\}$ . There is exactly one $k'\in\{1,\ldots,n\}$ with

\lfloor x\rfloor=\left\lfloor x+\frac{k'-1}{n}\right\rfloor\le x<\left\lfloor x+\frac{k'}{n}\right\rfloor=\lfloor x\rfloor+1.

By subtracting the same integer $\lfloor x\rfloor$ from inside the floor operations on the left and right sides of this inequality, it may be rewritten as

0=\left\lfloor \{x\}+\frac{k'-1}{n}\right\rfloor\le \{x\}<\left\lfloor \{x\}+\frac{k'}{n}\right\rfloor=1.

Therefore,

1-\frac{k'}{n}\le \{x\}<1-\frac{k'-1}{n} ,

and multiplying both sides by $n$ gives

n-k'\le n\, \{x\}<n-k'+1.

Now if the summation from Hermite's identity is split into two parts at index $k'$ , it becomes

{\begin{aligned}\sum _{k=0}^{n-1}\left\lfloor x+{\frac {k}{n}}\right\rfloor &=\sum _{k=0}^{k'-1}\lfloor x\rfloor +\sum _{k=k'}^{n-1}(\lfloor x\rfloor +1)=n\,\lfloor x\rfloor +n-k'\\[8pt]&=n\,\lfloor x\rfloor +\lfloor n\,\{x\}\rfloor =\left\lfloor n\,\lfloor x\rfloor +n\,\{x\}\right\rfloor =\lfloor nx\rfloor .\end{aligned}}

References

↑ Savchev, Svetoslav; Andreescu, Titu (2003), "12 Hermite's Identity", Mathematical Miniatures, New Mathematical Library, 43, Mathematical Association of America, pp. 41–44, ISBN 9780883856451 .
↑ Matsuoka, Yoshio (1964), "Classroom Notes: On a Proof of Hermite's Identity", The American Mathematical Monthly, 71 (10): 1115, doi:10.2307/2311413, MR 1533020 .

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[1] Savchev, Svetoslav; Andreescu, Titu (2003), "12 Hermite's Identity", Mathematical Miniatures, New Mathematical Library, 43, Mathematical Association of America, pp. 41–44, ISBN 9780883856451 .

[2] Matsuoka, Yoshio (1964), "Classroom Notes: On a Proof of Hermite's Identity", The American Mathematical Monthly, 71 (10): 1115, doi:10.2307/2311413, MR 1533020 .