An elementary extension of Hermite’s equality

Krassimir T. Atanassov
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 7, 2001, Number 2, Page 60
Full paper (PDF, 96 Kb)

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Krassimir T. Atanassov
CLBME – Bulg. Academy of Sci.,
P.O.Box 12, Sofia-1113, Bulgaria

Ch. Hermite has introduced the following well known equality for each positive real number $x$ and each natural number $n$

$[x] + [x + \frac{1}{n}] + [x + \frac{2}{n}] + \cdots + [x + \frac{n-1}{n}] = [nx].$

We shall extend it, proving that for each positive real number $x$ and each natural numbers $k$ and $n$

$[x] + [x + \frac{1}{n}] + [x + \frac{2}{n}] + \cdots + [x + \frac{kn - 1}{n}] = k[nx] + \frac{(k-1)k}{n}.$

Comtet, L. Advanced Combinatorics, D. Reidel Publ. Co. Dordrecht, 1974.
Vassilev M., Atanassov K., On Delanoy numbers, Annuaire de l’Universite de Sofia „St. Kliment Ohridski“, Faculte de Mathematiques et Informatique, Livre 1 – Mathematiques, Tome 81, 1987, 153-162.

Atanassov, Krassimir T. (2001). An elementary extension of Hermite’s equality. Notes on Number Theory and Discrete Mathematics, 7(2), 60.