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]

Proofs

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Proof by algebraic manipulation

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Split   into its integer part and fractional part,  . There is exactly one   with

 

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

 

Therefore,

 

and multiplying both sides by   gives

 

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

 

Proof using functions

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Consider the function

 

Then the identity is clearly equivalent to the statement   for all real  . But then we find,

 

Where in the last equality we use the fact that   for all integers  . But then   has period  . It then suffices to prove that   for all  . But in this case, the integral part of each summand in   is equal to 0. We deduce that the function is indeed 0 for all real inputs  .

References

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  1. ^ Savchev, Svetoslav; Andreescu, Titu (2003), "12 Hermite's Identity", Mathematical Miniatures, New Mathematical Library, vol. 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, JSTOR 2311413, MR 1533020.