I gurgled:
The expression (q;q)_oo persists in the reflection formula, hence my excitement at reexpressing it as a simple Theta_2.
DAWK! The equivalent and marginally nicer formula, inf pi 1/6 /===\ theta (--, q ) 1/24 | | n 1 3 q | | (1 - q ) = -----------------, | | sqrt(3) n = 1 has lurked for years as an immediate consequence of two neighboring identities in my Theta demo notebook, www.tweedledum.com/rwg/thet.gif, a scrollable bitmap too tall for IE. So the q-factorial reflection formula can be written 2 2 QF(- z, q ) QF(z - 1, q ) = 2 %pi ------ log(q) 4 %pi 1/3 (z - 1) z 2 theta (0, %e ) theta (---, q ) q (1 - q ) 2 1 3 ---------------------------------------------------------, 2 %pi ------ 2 %pi 1/6 log(q) 3 theta (---, q ) theta (%pi z, %e ) 1 3 1 and numerous clumsy expressions in the thet.gif Formulary (e.g., d45, d47, d102, d84, ...) can be simplified: 1 - n ----- n - 1 n %pi 1/3 2 n /===\ theta (---, q ) 3 theta (n z, q ) | | %pi j 1 3 1 | | theta (z + -----, q) = ----------------------------------------, | | 1 n %pi n/3 j = 0 theta (---, q ) 1 3 n - 1 /===\ | | %pi j | | theta (z + -----, q) = | | 2 n j = 0 1 - n ----- n %pi 1/3 2 %pi n theta (---, q ) 3 theta (n (z + ---), q ) 1 3 1 2 ------------------------------------------------, %pi n/3 theta (---, q ) 1 3 1 - n ----- n - 1 n %pi 1/3 2 n /===\ theta (---, q ) 3 theta (n z, q ) | | %pi j 1 3 4 | | theta (z + -----, q) = ----------------------------------------, | | 4 n %pi n/3 j = 0 theta (---, q ) 1 3 n - 1 /===\ | | %pi j | | theta (z + -----, q) = | | 3 n j = 0 1 - n ----- n %pi 1/3 2 %pi n theta (---, q ) 3 theta (n (z + ---), q ) 1 3 4 2 ------------------------------------------------, %pi n/3 theta (---, q ) 1 3 even ---- 2 /===\ | | %pi j even/2 %pi 1/3 %pi even/3 | | theta (-----, q) = theta (---, q ) theta (---, q ) | | 1 even 1 3 1 3 j = 1 2 even sqrt(-------------------------------------), even/2 + 1 theta (0, q) theta (0, q) 3 3 4 odd - 1 ------- 2 odd - 3 /===\ ------- | | %pi j 2 %pi 1/3 %pi odd/3 | | theta (-----, q) = theta (---, q ) theta (---, q ) | | 1 odd 1 3 1 3 j = 1 1 - odd ------- 4 3 sqrt(odd). --rwg