Krassimir T. Atanassov
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 7, 2001, Number 2, Page 60
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Krassimir T. Atanassov
CLBME – Bulg. Academy of Sci.,
P.O.Box 12, Sofia-1113, Bulgaria
Abstract
Ch. Hermite has introduced the following well known equality for each positive real number x and each natural number n
![\[[x] + [x + \frac{1}{n}] + [x + \frac{2}{n}] + \cdots + [x + \frac{n-1}{n}] = [nx].\]](https://nntdm.net/wp-content/ql-cache/quicklatex.com-f749c6c262eb96a23fa57d52c3c4588e_l3.png "Rendered by QuickLaTeX.com")
We shall extend it, proving that for each positive real number x and each natural numbers k and n
![\[[x] + [x + \frac{1}{n}] + [x + \frac{2}{n}] + \cdots + [x + \frac{kn - 1}{n}] = k[nx] + \frac{(k-1)k}{n}.\]](https://nntdm.net/wp-content/ql-cache/quicklatex.com-aa59e6d3be46568003d4ad62515bd885_l3.png "Rendered by QuickLaTeX.com")
References
- 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.
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Cite this paper
Atanassov, Krassimir T. (2001). An elementary extension of Hermite’s equality. Notes on Number Theory and Discrete Mathematics, 7(2), 60.
