How to perform this indefinite sum?

Question

The following indefinite sum with (CC[n]=1 for even n and CC[n]=i for odd n)

CC[n_] := (1/2 + I/2) (1 + I (-1)^(-1 + n))    

Sum[Exp[-\[Pi] \[Alpha]^2 (n2^2 + n3^2)] CC[n1 + n2] Conjugate[
CC[n1]] CC[n1 + n3] Conjugate[CC[n1 + n2 + n3]], {n1, 0, 
NN - 1}, {n2, -Infinity, Infinity}, {n3, -Infinity, Infinity}]

doesn't seem to work. Surely there is a better method?

Answer 1

Tell MMA additional assumptions

summand = 
   Exp[-\[Pi] \[Alpha]^2 (n2^2 + n3^2)] CC[n1 + n2] Conjugate[
   CC[n1]] CC[n1 + n3] Conjugate[CC[n1 + n2 + n3]] // 
   FullSimplify[#, 
   n1 \[Element] Integers && n2 \[Element] Integers && 
   n3 \[Element] Integers] &

Sum[summand, {n1, 0, NN - 1}, {n2, -Infinity, 
   Infinity}, {n3, -Infinity, Infinity}] /. NN -> 2

(*    1/8 (8 EllipticTheta[3, 0, E^(-\[Pi] \[Alpha]^2)]^2 + 
      16 EllipticTheta[3, 0, E^(-\[Pi] \[Alpha]^2)] EllipticTheta[4, 0, 
      E^(-\[Pi] \[Alpha]^2)] - 
     8 EllipticTheta[4, 0, E^(-\[Pi] \[Alpha]^2)]^2)    *)