I hadn't realized how directly connected they are to the Rogers-Ramanujan identities: If f(q):=(q,q^4;q^5)_oo and g(g):=(q^2,q^3;q^5)_OO, the reciprocals of the g.f.s for the +-1 (resp +-2) mod 5 partitions, then 5 5 f(q) 1 2 (q ;q )_oo ---- = -------- = f(q) --------- g(q) q (q;q)_oo 1 + -------- 2 q 1 + ------- 3 q 1 + ----- ... which appears to come out an <algebraic #>/q^(1/5) when q = e^(pi r), r a quadratic surd (in the left halfplane) "by an appeal to the theory of elliptic theta functions" (G. Andrews: The Theory of Partitions, p105 :-). Ramanujan sent Hardy the cases q = e^(-2 pi) and - e^-pi, which presumably give the nicest algebraics, f(e^-2 pi) pi 3 pi 2 pi/5 ---------- = 4 sin -- sin ---- e , g(e^-2 pi) 20 20 whereas the nicest cf is presumably f(e^-pi) sqrt(40 + 18 sqrt(5) - 25 5^(1/4) - 11 5^(3/4)) - sqrt(36 + 18 sqrt(5) - 25 5^(1/4) - 11 5^(3/4)) pi/5 -------- = ------------------------------------------------------------------------------------------------- e . g(e^-pi) 2 ~ 0.958650181596. Now we know why Ramanujan didn't send Hardy this one. I'm really interested in how to get these theta special values (other than by numerical searching), but Andrews seems to cite only combinatoric references rather than (theta) number-theoretic. In particular, where do I learn that f and g can be evaluated separately, e.g. here in terms of e^(pi/60) because -pi -pi - pi/6 10 f(e ) g(e ) = e sqrt ------------------ . 5^3/4 - 5^(1/4) +2 Other forms of f and g are n (5n-3) n/2 Sum (-) q 1 Theta_4(7 i log(q)/4, q^(5/2)) n f(q) = ----------- = - q ------------------------------ = -------------------- , n^2 (q^5;q^5)_oo (q^5;q^5)_oo q Sum ------- n>0 (q;q)_n the "gofigurate number theorem", and n (5n-1) n/2 Sum (-) q 1 2 Theta_4(9 i log(q)/4, q^(5/2)) n g(q) = ------------ = - q ------------------------------ = -------------------- . (n+1) n (q^5;q^5)_oo (q^5;q^5)_oo q Sum ------- n>0 (q;q)_n f(q) and g(q) become Theta_1(pi/5, Q) under Jacobi's transformation. If we can express these in terms of Theta_s(0,p) ("Theta constants") we'd have a nearly automatic source of special values because the Theta constants all come out as etas, at which we're pretty good. I can do Theta_1(pi/10, Q), but so far no luck with pi/5. --rwg