Have to admit I never looked into other than the *-null identities. Update on Somos' file, it's now at http://eta.math.georgetown.edu/ Currently at 6277(!) identities, many together with various equivalent in form of "known" functions such as Ramanujan's and Borwein^2's "cubic" a, b, and c. I STRONGLY suggest to consult this phantastic resource. A further Wikipedia page giving various neat identities is http://en.wikipedia.org/wiki/Ramanujan%E2%80%93Sato_series A book that since appeared is Köhler's "Eta Products and Theta Series Identities", see http://www.springer.com/gp/book/9783642161513 (can email pdf for "try before buy"). Best regards, jj P.S.: I met Jon Borwein a few years ago and asked whether there is any chance of a new edition of "Pi and the AGM", his answer was (sadly) a rather resounding "no". * rwg <rwg@sdf.org> [Jun 06. 2015 15:25]:
On 2009-07-05 01:35, Joerg Arndt wrote:
Hi, [not on the list, as unsure whether people \notin \{jj,rwg\} appreciate this stuff] List restored. Some of them do. This is just to remark that http://en.wikipedia.org/wiki/Dedekind_eta_function#Special_values gives the peculiar
DedekindEta[\[Tau]] == E^((I \[Pi] \[Tau])/12) EllipticTheta[3, 1/2 \[Pi] (1 + \[Tau]), E^(3 I \[Pi] \[Tau])]
instead of
DedekindEta[\[Tau]] == EllipticTheta[1, \[Pi]/3, (E^(2 I \[Pi] \[Tau]))^(1/6)]/Sqrt[3]
As a theta identity, this isn't obviously (to me) a quarter period or quarter quasiperiod identity, nor an imaginary transformation. The only thing I can guess is some sort of modular transformation of the eta. ? Which would suggest we add to our theta identities the modular eta relations, written as thetas? Will FullSimplify do these in our lifetime? --rwg
* rwg@sdf.lonestar.org <rwg@sdf.lonestar.org> [Feb 16. 2009 19:33]:
[...]
Just used a few brain cycles on the equality between third and last in this one:
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
via relations 29.2-26a ... 29.2-26c on page 617 of the fxtbook (version == 2009-June-30) this is (warning: partition eta strikes again!) Somos' relation t18_5_18 = +1*u1^3*u6*u9 +3*q*u1*u2*u3*u18^2 -1*u2^3*u3*u9 ; (line 2278) of his file eta07.gp.
Somos' file contains 250 such relations involving (a subset of) u1, u2, u3, u6, u9, u18, and >4k relations in total. So it may be feasible via some machinery to turn these into relations involving thetas (at least theta[*](z=0,q)). If so, _lots_ of relations would be obtained.
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
All trivially verified via the referenced relations, finding them may well be nontrivial.
--rwg
cheers, jj