Ed Pegg had me snooping in http://functions.wolfram.com/EllipticFunctions/DedekindEta/ where I was royally shocked to find an (unattributed) evaluation of eta(e^(-2 pi)). Which Mma 6.0 knows! This opens the floodgates, permitting closed forms for eta(e^(-r pi)) for *all* positive rational r. [...] (I just noticed %pi 1/3 2 2 2 theta'' (---, q ) eta (q ) 1 3 theta''' (0, q) = ------------------------------.) 1 sqrt(3)
Do we agree with Rich that the simplifier should prefer theta''(pi/3) to theta'''(0)?
I'm leaning the other way, because of block([%piargs:all,fancy_display:false], print(intosum(niceindices(subst([q=1/q,q=1/q],thetaderiv[s](z,n,q,k))))),0)$ k ==== Stirling_s1(k, i) thetaderiv (z, n + 2 i, q) \ s
--------------------------------------------, / i k k ==== 4 (- 1) q i = 0
i.e., d/dq ~ d^2/dz^2 . So you can completely eliminate k "dq"s at a cost of 2k "dz"s, but not vice versa, where you wind up with Floor(n/2) "dq"s, and a possible leftover dz. The additional k terms are a bit unflavorful, but are a cost typical of kth derivatives, e.g. of f(q)*g(q).
So far, no luck finding eta(e^-(pi sqrt(2))) etc,
LatticeReduce just churned up 1 3 sqrt(gamma(-) gamma(-)) - 2 sqrt(2) %pi 8 8 eta(%e ) = -----------------------, 1/8 3/4 2 8 %pi 2 %pi 1/8 1 2 4 - ------- 7 sqrt(gamma(-) gamma(-) gamma(-)) sqrt(7) 7 7 7 eta(%e ) = -------------------------------------, 2 sqrt(2) %pi 4 %pi - ------- sqrt(7) eta(%e ) = 1/8 1/4 1 2 4 7 (sqrt(7) + 3) sqrt(gamma(-) gamma(-) gamma(-)) 7 7 7 ------------------------------------------------------, 1/8 4 2 %pi 1/8 3/2 1 3 gamma (-) - 2 sqrt(3) %pi 3 eta(%e ) = ----------------, 1/3 2 2 %pi 2 %pi 3/8 3/2 1 - ------- 3 gamma (-) sqrt(3) 3 eta(%e ) = ----------------, 1/3 2 2 %pi 4 %pi 3/8 1/4 3/2 1 - ------- 3 (sqrt(3) + 1) gamma (-) sqrt(3) 3 eta(%e ) = ---------------------------------, 1/8 1/4 2 2 8 %pi and might be coaxed into e^-(pi sqrt(n)) for any given positive integer, from which we can get pi <rational> sqrt(n) with the trivariate polynomials, at least for <rational> made of small primes. Automating this for eta|theta(0) simplification would make a fairly weird piece of nonrigorous (due to the LatticeReduce) computer algebra. Actually, the trivariate polynomials (for general eta(q)) are nonrigorous too, having been found with a Taylor series analog of LatticeReduce. But I dimly recall there being a theorem bounding the number of terms you need to check to guarantee equality. This pushes the luck frontier back to, e.g., e^-(pi sqrt(3/2)),
and eta' of anything besides e^-(2 pi). Sadly. (One more of the latter would open another floodgate.)
--rwg