Is there any formula for this sum of power of positive integers? I wonder if there is any formula for this sum.
$$k^\gamma+(k-1)^\gamma+\cdots+1^\gamma,$$
where $k$ is positive integer and $\gamma\in(0,1)$. And how about $\gamma<0$? Or is there any known asymptotic when $k$ tends to infinity?
 A: The sum gives $\sum _{j=1}^k j^\gamma = H_k^{(-\gamma)}$ which is the harmonic number of order $\gamma.$
An excellent approximation I have found is
$$\sum _{j=1}^k j^{\gamma}\approx \left(\frac{\gamma}{12 k}+\frac{k}{\gamma+1} + \frac{1}{2}\right) k^{\gamma}+\frac{1}{2} \gamma \log (2 \pi )-\frac{1}{2}$$
for $\gamma=0.5$ I got the following results. It is an asymptotic approximation, so it works better for large $k$. For negative $\gamma$ the approximation works fine if $|\gamma|<0.5$
$$
\begin{array}{r|r|r}
k & H_k^{(-\frac12)}& \textit{approximation}\\
\hline
 1 & 1 & 1.1678 \\
 11 & 25.7849 & 25.9523 \\
 21 & 66.2486 & 66.4159 \\
 31 & 117.651 & 117.818 \\
 41 & 178.019 & 178.186 \\
 51 & 246.177 & 246.345 \\
 61 & 321.319 & 321.487 \\
 71 & 402.848 & 403.015 \\
 81 & 490.297 & 490.464 \\
 91 & 583.289 & 583.457 \\
 101 & 681.513 & 681.68
\end{array}
$$
A: This seems to be studied here:
The generalization of Faulhaber's formula to sums of non-integral powers
(McGowna. Parks - Journal of Mathematical Analysis and Applications - v 330, 1, 1 June 2007)
There it's proved that
$$ (a+1) \sum_{n=1}^N n^a = N^\gamma F_a(N) + (a+1) \zeta(-a)+O(N^{-\beta})$$
where $a\ge -1$ and non integer (actually it's also valid for complex $a$ with real part greater than $1$), $\beta = m -a$, $m=\lfloor a+1 \rfloor $, $\gamma = -(m-a)$ and
$$F_a(N)=N^{m+1} + \sum_{k=1}^m (-1)^k {a+1 \choose k}B_k N^{m-k+1}$$
with $B_k$ denoting Bernoulli numbers.
