Bound for a double sum and power series

Question

I am looking for a bound for the sum of of the following form:

$$\sum_{1 \leq l <k \leq n} (k-l)^{\gamma}$$ and $\gamma >1$. Does somebody knows what would be an upper bound for such sum involving preferably only a sum with summation over $k$ and involving parameter $\gamma?$

Answer 1

The given sum can be expressed as $$ \sum_{1 \leq l <k \leq n} (k-l)^{\gamma}=\sum_{l=1}^{n-1}\sum_{k=1}^{l-1}{k^{\gamma}}=\sum_{k=1}^{n-1}{(n-k)k^{\gamma}}=n\sum_{k=1}^{n-1}{k^{\gamma}}-\sum_{k=1}^{n-1}k^{\gamma+1} $$ The last two sums can be bounded by definite integrals in the usual way.