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:.

: $\sum_^\left\lfloor x+\frac\right\rfloor=\lfloor nx\rfloor .$

Proof

Split $x$ into its integer part and fractional part, $x=\lfloor x\rfloor+\$. There is exactly one $k'\in\$ with
:$\lfloor x\rfloor=\left\lfloor x+\frac\right\rfloor\le x<\left\lfloor x+\frac\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

:$0=\left\lfloor \+\frac\right\rfloor\le \<\left\lfloor \+\frac\right\rfloor=1.$

Therefore,

:$1-\frac\le \<1-\frac ,$

and multiplying both sides by $n$ gives

:n-k'\le n\, \

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

:
\begin
\sum_^\left\lfloor x+\frac\right\rfloor
& =\sum_^ \lfloor x\rfloor+\sum_^ (\lfloor x\rfloor+1)=n\, \lfloor x\rfloor+n-k' \\ pt& =n\, \lfloor x\rfloor+\lfloor n\,\\rfloor=\left\lfloor n\, \lfloor x\rfloor+n\, \ \right\rfloor=\lfloor nx\rfloor.
\end

Alternate proof

Consider the function

:$f(x) = \lfloor x \rfloor + \left\lfloor x + \frac \right\rfloor + \ldots + \left\lfloor x + \frac \right\rfloor - \lfloor nx \rfloor$

Then the identity is clearly equivalent to the statement $f(x) = 0$ for all real $x$. But then we find,

:$f\left(x + \frac \right) = \left\lfloor x + \frac \right\rfloor + \left\lfloor x + \frac \right\rfloor + \ldots + \left\lfloor x + 1 \right\rfloor - \lfloor nx + 1 \rfloor = f(x)$

Where in the last equality we use the fact that $\lfloor x + p \rfloor = \lfloor x \rfloor + p$ for all integers $p$. But then $f$ has period $1/n$. It then suffices to prove that $f(x) = 0$ for all $x \in [0, 1/n)$. But in this case, the integral part of each summand in $f$ is equal to 0. We deduce that the function is indeed 0 for all real inputs $x$.

References

{{DEFAULTSORT:Hermite's Identity
Mathematical identities
Articles containing proofs