The Theta constants in terms of eta: {Equal[EllipticThetaPrime[1, 0, q], 2*(DedekindEta[-((Log[q^2]*I)/(2*Pi))])^3], Equal[EllipticTheta[2, 0, q], ((2*(DedekindEta[-((Log[q^4]*I)/(2* Pi))])^2)/(DedekindEta[-((Log[q^2]*I)/(2*Pi))]))], Equal[EllipticTheta[3, 0, q], (((DedekindEta[-((Log[q^2]*I)/(2* Pi))])^5)/((DedekindEta[-((Log[q]*I)/(2* Pi))])^2*(DedekindEta[-((Log[q^4]*I)/(2*Pi))])^2))], Equal[EllipticTheta[4, 0, q], (((DedekindEta[-((Log[q]*I)/(2*Pi))])^2)/(DedekindEta[-((Log[ q^2]*I)/(2*Pi))]))]} Actually, there are many other Theta constants, like ...Theta[...,π/3,q], Theta[...,π/4,q],... that can be done with etas. I didn't mention π/2 because things like In[522]:= FullSimplify@%519 Out[522]= EllipticTheta[1, \[Pi]/2, q] == EllipticTheta[2, 0, q] should autosimplify! In[523]:= Series[%, {q, 0, 69}] Out[523]= True Using the modular property of eta, plus the above, provides such futuristic benchmarks as EllipticThetaPrime[1, 0, E^(-(1/13) (I + 2 Sqrt[3]) \[Pi])] == ((-1)^(1/4) 3^( 3/8) (((11 - 4 I Sqrt[3]) (Sqrt[2] + Sqrt[3]))/((1 + Sqrt[3]) (2^( 3/4) + 2^(1/4) Sqrt[3])))^(3/4) Gamma[1/3]^(9/2))/( 8 2^(11/16) \[Pi]^3), EllipticTheta[2, 0, E^(-(1/13) (I + 2 Sqrt[3]) \[Pi])] == ( 3^(1/8) Sqrt[1 + Sqrt[2]] ((11 - 4 I Sqrt[3]) (Sqrt[2] + Sqrt[3]))^( 1/4) (98 + 40 Sqrt[6])^(1/16) Gamma[1/3]^(3/2))/( 2 2^(23/48) Sqrt[1 + Sqrt[3]] \[Pi]), EllipticThetaPrime[1, 0, E^(-(2/13) (I + 2 Sqrt[3]) \[Pi])] == ((-1)^(1/8) 3^( 3/8) ((1 + Sqrt[2]) (11 - 4 I Sqrt[3]) (Sqrt[2] + Sqrt[3]))^(3/4) Gamma[1/3]^(9/2))/(32 2^(1/16) (1 + Sqrt[3])^(9/8) \[Pi]^3), EllipticTheta[4, 0, E^(-(2/13) (I + 2 Sqrt[3]) \[Pi])] == ((-3)^( 1/8) ((11 - 4 I Sqrt[3]) (Sqrt[2] + Sqrt[3]))^(1/4) Gamma[1/3]^( 3/2))/(2 2^(5/48) (1 + Sqrt[2])^(1/4) (1 + Sqrt[3])^(1/8) Sqrt[ 2^(3/4) + 2^(1/4) Sqrt[3]] \[Pi]) —Bill On Sat, Jan 26, 2019 at 12:36 AM Bill Gosper <billgosper@gmail.com> wrote:
(Probably repeating myself, but) In[86]:= eta124
Out[86]= 0 == -DedekindEta[2 \[Tau]]^24 + DedekindEta[\[Tau]]^16 DedekindEta[4 \[Tau]]^8 + 16 DedekindEta[\[Tau]]^8 DedekindEta[4 \[Tau]]^16
E.g., In[81]:= eta124 /. \[Tau] -> I/2/\[Sqrt]3
Out[81]= 0 == -DedekindEta[I/Sqrt[3]]^24 + DedekindEta[I/(2 Sqrt[3])]^16 DedekindEta[(2 I)/Sqrt[3]]^8 + 16 DedekindEta[I/(2 Sqrt[3])]^8 DedekindEta[(2 I)/Sqrt[3]]^16
In[84]:= FullSimplify@%81
Out[84]= DedekindEta[I/Sqrt[3]]^24 == DedekindEta[I/(2 Sqrt[3])]^8 DedekindEta[(2 I)/Sqrt[ 3]]^8 (DedekindEta[I/(2 Sqrt[3])]^8 + 16 DedekindEta[(2 I)/Sqrt[3]]^8)
12.0 can't even do this case, let alone symbolic 1-2-4.
In[85]:= N@%
Out[85]= True In[91]:= %84 /. etavals
Out[91]= (19683 Gamma[1/3]^36)/(4294967296 \[Pi]^24) == ( 729 Gamma[1/ 3]^24 ((27 Gamma[1/3]^12)/(512 (1 + Sqrt[3])^2 \[Pi]^8) + ( 27 (1 + Sqrt[3])^2 Gamma[1/3]^12)/(2048 \[Pi]^8)))/( 16777216 \[Pi]^16)
In[92]:= FullSimplify@%
Out[92]= True
Written as infinite products, 1-2-4 is Jacobi's Æquatio Identica Satis Abstrusa.
Besides 1-2-4, there are relations for 1-3-9, 1-4-16, 1-5-25, ... There is even 1-2-3: DedekindEta[\[Tau]]^60 DedekindEta[2 \[Tau]]^24 - DedekindEta[\[Tau]]^72 DedekindEta[3 \[Tau]]^12 - 12 DedekindEta[\[Tau]]^48 DedekindEta[2 \[Tau]]^24 DedekindEta[ 3 \[Tau]]^12 - 119168 DedekindEta[\[Tau]]^24 DedekindEta[2 \[Tau]]^48 DedekindEta[ 3 \[Tau]]^12 - 16777216 DedekindEta[2 \[Tau]]^72 DedekindEta[3 \[Tau]]^12 + 196830 DedekindEta[\[Tau]]^36 DedekindEta[2 \[Tau]]^24 DedekindEta[ 3 \[Tau]]^24 + 19131876 DedekindEta[\[Tau]]^24 DedekindEta[2 \[Tau]]^24 DedekindEta[ 3 \[Tau]]^36 + 387420489 DedekindEta[\[Tau]]^12 DedekindEta[ 2 \[Tau]]^24 DedekindEta[3 \[Tau]]^48
and presumably a-b-c for arbitrary distinct positive rationals. With Macsyma I once computed a big list of these, which seems now to break off in the middle of 1-4-5. Anyway, this claims that, e.g., DedekindEta[22I/7] is some monstrous algebraic number times DedekindEta[I] = Gamma[1/4]/(2 \[Pi]^(3/4)). —Bill The reason I nag is that nearly everything is made out of Etas—Theta constants, elliptic Integrals, moduli, lambda*, Klein's invariant, . . ., and then there is Eta' (another case where a function's derivative is hairier than the function).