From: rwg@sdf.lonestar.org [mailto:rwg@sdf.lonestar.org] Sent: Wed 7/9/2008 9:22 AM To: math-fun@mailman.xmission.com Cc: "rwg@sdf.lonestar.orgrwg"@sdf.lonestar.org Subject: Theta function halfangles
(I promised Cheny Xu, proprietor of Jade Palace Restaurant,
www.jadepalace.us
that I'd solve a problem with the help of some fish he fed me last night.)
It's not obvious how to extract a tractable halfangle formula from the many double angle formulae in W&W. After you eliminate the three undesired species, the resultant polynomial is too hairy. Actually, it would probably simplify if Macsyma or Mma only knew how, but I can't even get Mma 6.0 to EllipticTheta[1, Pi, q] -> 0 ! (Hey guys, need a programmer?)
Fortunately, I (re?)conjectured a family of theta determinant identities, http://www.tweedledum.com/rwg/thet.PNG (too tall for IE 7), arxiv.org/pdf/math/0703470, the first few (including the one used here) proved by Rich (and Hilarie?). These are nice when actually typeset:
x theta (-, q) = 1 2
3 3 3 - theta (0, q) theta (x, q) + theta (0, q) theta (x, q) - theta (0, q) theta (x, q) 4 4 3 3 2 2 1/4 (-----------------------------------------------------------------------------------) , 2
x theta (-, q) = 2 2
3 3 3 - theta (0, q) theta (x, q) + theta (0, q) theta (x, q) + theta (0, q) theta (x, q) 4 4 3 3 2 2 1/4 (-----------------------------------------------------------------------------------) , 2
x theta (-, q) = 3 2
3 3 3 theta (0, q) theta (x, q) + theta (0, q) theta (x, q) + theta (0, q) theta (x, q) 4 4 3 3 2 2 1/4 (---------------------------------------------------------------------------------) , 2
x theta (-, q) = 4 2
3 3 3 theta (0, q) theta (x, q) + theta (0, q) theta (x, q) - theta (0, q) theta (x, q) 4 4 3 3 2 2 1/4 (---------------------------------------------------------------------------------) . 2
E.g., pi pi theta (--, q) = theta (--, q) = 1 4 2 4
2 2 theta (0, q) theta (0, q) (theta (0, q) - theta (0, q)) 3 4 3 4 1/4 (-------------------------------------------------------) , 2
pi pi theta (--, q) = theta (--, q) = 3 4 4 4
2 2 theta (0, q) theta (0, q) (theta (0, q) + theta (0, q)) 3 4 4 3 1/4 (-------------------------------------------------------) . 2
But, duh, %pi %pi 4 theta (---, q) = theta (---, q) = theta (0, q ) 4 4 3 4 4
Finally, this must be in Tannery & Molk, but I don't recall seeing a Lambert type series come out as a pure theta instead of a log derivative theta:
inf 2 ==== j 2 j - 1 theta (0, q) - 1 \ (- 1) q 4 > --------------- = ----------------. / 2 j - 1 4 ==== q + 1 j = 1
There seems to be a lot of these. theta (0, q) 2 log(-------------------) inf 1/4 ==== n n 2 theta (0, q) q \ (- 1) q 3 > ---------- = ------------------------. / n 2 ==== n (q + 1) n = 1
--rwg