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:

Proofs

Proof by algebraic manipulation

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

Proof using functions

Consider the function Then the identity is clearly equivalent to the statement f(x) = 0 for all real x. But then we find, Where in the last equality we use the fact that for all integers p. But then f has period 1/n. It then suffices to prove that f(x) = 0 for all. 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.