It is known that exists recurrent formula for counting $$S_m(n)=\sum_{i=1}^{n}i^m,S_0(n)=n,m>0$$ I am intrested if such formulas exists for $m<0$
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This MSE link treats the case $m=-2$. – Marko Riedel Apr 25 '14 at 22:13
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Newton's identities relate via recurrence formulae the powers sums $\sum_{i=1}^{n} x_{i}^{j}$ to the elementary symmetric functions in the variables $x_{i},$ which are (up to a uniquely determined sign in each case) the coefficients of the powers of $t$ in the polynomial $\prod_{i=1}^{n} (t-x_{i}).$ You can then evaluate the $m$-th power sum setting each $x_{i} = i^{-1}$ to obtain one answer to your question.
Geoff Robinson
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Can you show how to apply Newton identities for example in case $m=-1,-2,...$ – Adi Dani Jan 10 '13 at 13:20
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$H_{n,m}=\sum_{k=1}^n \frac{1}{k^m}=\sum_{k=1}^n k^{-m}=S_{-m}(n),m>0$ is generalized harmonic number – Adi Dani Jan 10 '13 at 13:45
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Newton's identities work for any values of the variables, since they are polynomial identities. That is why you can work with $x_{i} = i^{-1}$ instead of $x_{i} = i,$ and then you have taken care of negative values of $m.$ – Geoff Robinson Jan 10 '13 at 15:32