In[150]:= EllipticTheta[1, π/3, q^(1/6)]/√3/q^(1/24) // theta2product Out[150]= Product[(1 - q^(n/3))*(1 + q^(n/3) + q^((2*n)/3)), {n, ∞}] (I.e., (q;q)_∞.) In[149]:= EllipticTheta[1, π/3, q^(1/6)]/√3/q^(1/24) // theta2sum Out[149]= (2*Sum[(-1)^n*q^((1/6)*(1/2 + n)^2)*Sin[(1/3)*(1 + 2*n)*Pi], {n, 0, ∞}])/ (√3*q^(1/24)) Group terms in threes: In[152]:= gruple[%149, 3] Out[152]= (1/(Sqrt[3]*q^(1/24)))*2* Sum[ (-1)^(-3 + 3*n)*q^((1/6)*(-(5/2) + 3*n)^2)*Sin[(1/3)*(1 + 2*(-3 + 3*n))*π] + (-1)^(-2 + 3*n)*q^((1/6)*(-(3/2) + 3*n)^2)*Sin[(1/3)*(1 + 2*(-2 + 3*n))*π] + (-1)^(-1 + 3*n)*q^((1/6)*(-(1/2) + 3*n)^2)*Sin[(1/3)*(1 + 2*(-1 + 3*n))*π], {n,∞}] Remind Mathematica that n is an integer: In[154]:= Assuming[n \[Element] Integers, MapAt[Simplify /@ # &, %, {-1, 1}]] Out[154]= (2*Sum[(1/2)*(-1)^n*√3*q^((1/24)*(1 - 6*n)^2) - (1/2)*(-1)^n*√3*q^((1/24)*(5 - 6*n)^2), {n, ∞}])/ (√3*q^(1/24)) In[160]:= ExpandAll@intosum@%154 Out[160]= Sum[(-1)^(1 + n)*q^(1 - (5*n)/2 + (3*n^2)/2) + (-1)^n*q^(-(n/2) + (3*n^2)/2), {n, ∞}] —rwg Anybody want theta2product, theta2sum, gruple, intosum? Let me know. It's actually a little disappointing that Mathematica doesn't automatically revert back to EllipticTheta! When that day comes we'll need to Inactive the Sum and Product.