From: rwg@sdf.lonestar.org [mailto:rwg@sdf.lonestar.org] Sent: Wed 7/9/2008 9:22 AM To: math-fun@mailman.xmission.com [...] 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. [...]
George Andrews kindly informs me that E. Grosswald ascribes this to Jacobi and his counting of representations as sums of two squares. It generalizes to inf 2 ==== n theta (0, q) theta (x, q) theta (x, q) \ q sin(2 n x) cot(x) 4 2 3
------------- = ------ - --------------------------------------. / n 4 4 theta (x, q) theta (x, q) ==== q + 1 1 4 n = 1
The reason for no thetaprimes is that it's the derivative of csc(x) theta (x, q) 1 log(-------------------) inf 1/4 ==== n 2 q theta (x, q) \ q cos(2 n x) 4 > ------------- = ------------------------, / n 2 ==== n (q + 1) n = 1 and the derivative of a quotient of two thetas comes out in thetas. Theta quotients come from subtracting variations on %pi 1/6 2 theta (---, q ) sin(z) 1 3 inf log(--------------------------) ==== n sqrt(3) theta (z, sqrt(q)) \ q cos(2 n z) 1 log(q) > ------------- = ------------------------------- + ------ / n 2 24 ==== n (1 - q ) n = 1 and %pi 1/3 theta (---, q ) 1 3 inf log(--------------------) ==== n sqrt(3) theta (z, q) \ q cos(2 n z) 4 log(q) > ------------- = ------------------------- - ------, / 2 n 2 24 ==== n (1 - q ) n = 1 which I dredged out of W&W. This gives us, e.g., inf ==== n \ q log(q) %pi 1/6 log(3) > ---------- = ------ - log(theta (---, q )) + ------, / n 24 1 3 2 ==== n (1 - q ) n = 1 but still nothing about plain old inf ==== n \ q > ------, / n ==== q - 1 n = 1 with which we could sum the reciprocal Fibonacci numbers. Here's another pure theta sum: inf ==== 2 n - 1 \ q sin((2 n - 1) z)
------------------------- = / 2 n - 1 ==== 1 - q n = 1
2 z z theta (0, sqrt(q)) theta (-, sqrt(q)) theta (-, sqrt(q)) 2 3 2 4 2 csc(z) -------------------------------------------------------- - ------ z z 4 8 theta (-, sqrt(q)) theta (-, sqrt(q)) 1 2 2 2 --rwg