I have faced the following sum in my calculation:
$$\sum_{l_1=0}^{N_1-1}\sum_{l_2=0}^{N_2-1}\sum_{l_3=0}^{N_3-1}\exp{[-ia(q_xl_1+q_yl_2+q_zl_3)]}\times\exp{-i[q\beta\times\text{e}^{ia[k_xl_1+k_yl_2+k_zl_3]}]}$$
where, all the quantities $q\beta$, $a(q_x+q_y+q_z)$ and $(k_x+k_y+k_z)a$ are made dimensionless.
This summation makes my whole calculation very complex and I want to find out the closed form result of this summation and accordingly what I tried to write the above summation in the following way: $$\sum_{l_1,l_2,l_3}\exp{-ia(q_xl_1+q_yl_2+q_zl_3)}\times\sum_{l_1',l_2',l_3'}\exp{-i[q\beta\times\text{e}^{ia[k_xl_1'+k_yl_2'+k_zl_3']}]}\delta _{l_1l_1'}\delta_{l_2,l_2'}\delta_{l_3,l_3'}$$
The summation in the first part of the above expression can be evaluated following
$$\sum_{s=0}^{N-1}\exp{(-ik_xas)}=N \exp{[-i\frac{(N-1)k_x a}{2}]}\times \frac{sinc{(\frac{Nk_x a}{2}})}{sinc{(\frac{k_x a}{2}})}$$
but the closed form answer of the second part is not found.
I also searched in "Table of Integrals by Gradshteyn and Ryzhick" but I could not find the exact thing.
However, I am confused with one thing that for the 2nd part can I write the following $$\sum_{l'}f(l')\delta_{ll'}=f(l)$$
Would you kindly suggest me any way or any relevant documents from where I can get some help to find out the desired answer