Tentative quadrivariate 𝜗₁ identity: Out[742]= EllipticTheta[1, -X + Y, q] EllipticTheta[1, X + Y, q] EllipticTheta[ 1, W - Z, q] EllipticTheta[1, W + Z, q] - EllipticTheta[1, W - Y, q] EllipticTheta[1, W + Y, q] EllipticTheta[ 1, -X + Z, q] EllipticTheta[1, X + Z, q] + EllipticTheta[1, W - X, q] EllipticTheta[1, W + X, q] EllipticTheta[ 1, -Y + Z, q] EllipticTheta[1, Y + Z, q] (= 0?) Testing to 0ᵗʰ order: In[743]:= Series[%742 /. {X -> x*q, Y -> y*q, Z -> z*q, W -> w*q}, {q, 0, 0}] Out[743]= O[q]^8 Nice. (Too nice? I requested essentially no terms.) OK, try 7ᵗʰ: In[745]:= Series[%742 /. {X -> x*q, Y -> y*q, Z -> z*q, W -> w*q}, {q, 0, 7}] Out[745]= O[q]^8 Why no improvement? Trying for 8ᵗʰ, In[747]:= Series[%742 /. {X -> x*q, Y -> y*q, Z -> z*q, W -> w*q}, {q, 0, 8}] // FunctionExpand hits a lacuna in Mma's 𝜗' knowledge: ((1/2 (-x+y)^2 (EllipticThetaPrime^(0,1,0))[1,0,0]+ <big mess>...)) So let's try random exact numerics: In[759]:= Table[v -> RandomInteger[{-9, 9}, 2].{1, I}, {v, {W, X, Y, Z}}] Out[759]= {W -> -3 + 6 I, X -> -7 I, Y -> 6 - 6 I, Z -> -7 + 9 I} In[760]:= Table[v -> RandomInteger[{-9, 9}, 2].{1, I}/9/Sqrt[2], {v, {q}}] Out[760]= {q -> -((5 I)/(9 Sqrt[2]))} In[761]:= %742 /. %% /. % Out[761]= EllipticTheta[1, -13 + 15 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -3 - I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -3 + 13 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -1 + 3 I, -((5 I)/(9 Sqrt[2]))] - EllipticTheta[1, -9 + 12 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -7 + 2 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[ 1, -7 + 16 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 3, -((5 I)/(9 Sqrt[2]))] + EllipticTheta[1, -10 + 15 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 4 - 3 I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 6 + I, -((5 I)/(9 Sqrt[2]))] EllipticTheta[1, 6 - 13 I, -((5 I)/(9 Sqrt[2]))] In[762]:= N[%] Out[762]= -2.20279*10^173 + 1.54582*10^173 I I was kinda hoping for 0. How discouraging. But wait! In[764]:= N[%%%, 999] During evaluation of In[764]:= N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating EllipticTheta[1,-13+15 I,-((5 I)/(9 Sqrt[2]))] EllipticTheta[1,-3-I,-((5 I)/(9 Sqrt[2]))] EllipticTheta[1,-3+13 I,-((5 I)/(9 Sqrt[2]))] EllipticTheta[1,-1+3 I,-((5 I)/(9 Sqrt[2]))]-<<1>><<1>>EllipticTheta[1,-10+15 I,-((5 I)/(9 Sqrt[2]))] <<13>>[<<1>>] <<1>> <<1>>. >> Out[764]= 0.*10^-856 + 0.*10^-856 I How can people regard symbolic math as more exotic than numerics? --rwg