Fourier transformation of summation of bosonic operators with alternate signs.

Question

The sum of the following Bosonic operators: $$ \sum_{<ij>}a_{i}^{+}a_{j}+a_{j}^{+ }a_{i} $$ where i is the lattice site index(i=1 to N, it has periodic boundary condition, $$ r_{i+N}=r_{i}$$, j: nearest neighbour, and it can be rewritten as: $$ (1/2) \times \sum_{i,i+\delta}a_{i}^{+}a_{i+\delta}+a_{i+\delta}^{+}a_{i} $$ The Fourier Transformation is defined as $$ a_{i}^{+}=\frac{1}{\sqrt{N}}\sum_{k}e^{-i.\vec k. \vec r_{i}}a_{k}^{+}\\ a_{i}=\frac{1}{\sqrt{N}}\sum_{k}e^{i.\vec k. \vec r_{i}}a_{k} $$ The rest of the calculation is straightforward and it gives us $$ \frac{1}{2}\sum_{\delta,k}(e^{i.\vec k.\vec \delta} + e^{-i.\vec k.\vec \delta})a_{k}^{+}a_{k} $$ The question is, How to perform the Fourier transformation of the series: $$ \sum_{<ij>}(-1)^{(i+1)} (a_{i}^{+}a_{j}+a_{j}^{+}a_{i})$$ and what will be the answer? I am getting 0. Also, any kind of reference to related mathematical texts will be appreciated. Regards.