Hermite's identity

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This article is not about Hermite's cotangent 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]

Proof[edit]

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

References[edit]

  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 .