A couple of years ago I restated an almost forgotten identity equivalent to
%pi %pi 6 theta (---, q) = theta (---, q) = sqrt(3) eta(q ) 2 6 1 3
3 2 6 theta (0, q ) theta (0, q ) 2 4 1/3 = sqrt(3) (---------------------------) , 2
in this form as special values of Theta(pi/n) in terms of eta or equivalently the "theta constants" theta[s](0,q^k). What is the analogous value with the lhs 1 and 2 switched? 2 2 12 %pi %pi eta (q ) eta(q ) theta (---, q) = theta (---, q) = ----------------- 2 3 1 6 4 6 eta(q ) eta(q )
3 9 theta (0, q ) theta (0, q) - 3 theta (0, q ) 2 2 1/3 2 2 = theta (0, q ) (---------------) = ------------------------------. 4 6 2 2 theta (0, q ) 4
This is a rich area of nonobvious(?) identities, e.g.,
2 6 6 theta (0, q) theta (0, q ) theta (0, q ) 2 2 3 2 2 2 3 = theta (0, q ) theta (0, q ) theta (0, q ), 2 3 2 or equivalently 2 6 theta (0, q) theta (0, q) theta (0, q ) 3 4 4
2 2 3 3 = theta (0, q ) theta (0, q ) theta (0, q ) 4 3 4
The other two thetas at pi/3|6 are 3 3 2 3 theta (0, q) theta (0, q) theta (0, q ) pi 2 4 3 1/6 theta (--, q) = (---------------------------------------) 3 3 3 3 4 theta (0, q ) theta (0, q ) 2 4 9 4 2 6 3 theta (0, q ) - theta (0, q) pi eta(q) eta(q ) eta (q ) 3 3 = theta (--, - q) = ------------------------ = ------------------------------ 4 3 2 3 12 2 eta(q ) eta(q ) eta(q ) pi = theta (--, q), 4 6 and 3 theta (0, q ) 2 2 3 pi 4 1/3 eta (q ) eta(q ) theta (--, q) = theta (0, sqrt(q)) (-----------------) = ---------------- = 4 3 2 3/2 6 4 theta (0, q ) eta(q) eta(q ) 2 9 3 theta (0, q ) - theta (0, q) 4 4 pi ------------------------------ = theta (--, q). 2 3 6 I started with 2 2 pi 2 2 pi sqrt(theta (0, q) theta (--, q) + theta (0, q) theta (--, q)) pi 3 1 3 4 1 6 theta (--, q) = --------------------------------------------------------------- 3 6 theta (0, q) 2 from a determinant identity and got absolutely stuck trying for monomials. One thing that should've worked was that power series analog of LatticeReduce that I keep hyping. It's failure in both Macsyma and Mma nearly convinced me there weren't any monomials. I still don't know where I screwed up, although a streamlined version that only proposes constant coefficients, vs polynomials in q, would've settled things promptly. [Later] I expected expansion to q^24 or so would suffice, but to be "sure", chose q^99, w/o success. But q^199 just now succeeded in Mma, taking most of an hour (prior experiences would predict a few seconds), scaling by a 5.5 page rational number! (So a lot of time was wasted reducing huge fractions. You can turn that off in Macsyma with gcd:false.) Aha. I'm using the series for the log eta and trying to find the powers. The q^(k/24) multiplier of the eta contributes a k*log(q) term which both systems (mis)treat as constant. Knowing the logs will wash out anyway, I replaced them by 0, destroying the washout information. Replacing them instead by 1/q gets the time down to 58 secs(!) and needs q^(<=79). (But q^(>69).) --rwg