Calculate sum of series $\sum_{i=1}^{\infty} a_i a_{i + k}$ for any $k \in \mathbb{N}$

I'm trying to find closed form of the sum of the following series$$\sum_{n=1}^{\infty}\left[\rule{0pt}{5mm}\left(n + 1\right)^{\alpha} + \left(n - 1\right)^{\alpha} - 2n^{\alpha}\right]\left[\rule{0pt}{5mm}\left(n + 1 + k\right)^{\alpha} + \left(n + k - 1\right)^{\alpha}-2\left(n + k\right)^{\alpha}\right],$$where $\alpha \in \left(0, 2\right), k \in \mathbb{N}$.

• I've tried several approaches, tried to represent this sum as integral sum for some function, to make it telescopic, but haven't any success.

Will be very thankful for any help.