**Authors:** S. Maiti

The paper is focused to find important relations and identities over some summations for natural numbers such as $\displaystyle \sum_{\substack{i,j=1\\(i<j)}}^{n}ij,~\sum_{\substack{i,j,k=1\\(i<j<k)}}^{n}ijk,~\sum_{\substack{i,j,k,l=1 \\(i<j<k<l)}}^{n}ijkl,~\cdots $. These relations are believed to find applications in the various branches of number theory particularly in the proposed theorems of Maiti \cite{Maiti1,Maiti2,Maiti3,Maiti4,Maiti5} which help to represent the factorial $n!$ entirely new way and also help to exhibit the $n$th partial sum of the general harmonic series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{a+(n-1) b}$ and its particular cases.

**Comments:** 5 Pages.

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[v1] 2012-07-24 13:15:05

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