For n in {7, 11, 19, 43, 67, 163}, 2*%pi*sqrt(eta(%e^-(%pi*sqrt(n)))^8+16*eta(%e^-(4*%pi*sqrt(n)))^8)*n = product(gamma(k/n)^jacobi(k,n),k,1,n-1) n - 1 /===\ 8 1 8 1 | | jacobi(k, n) k 2 sqrt(eta (-----------) + 16 eta (-------------)) n pi = | | gamma (-) sqrt(n) pi 4 sqrt(n) pi | | n e e k = 1 (actually derived from the conjectural J^(1/3) result, using eta(%i*q)=eipi(-5/48)*eta(q^4)*(4*(eta(q^8)/eta(q^2))^2+%i*(eta(q^2)/eta(q^8))^2)^(1/4) 2 8 2 2 4 4 eta (q ) i eta (q ) 1/4 eta(q ) (---------- + ----------) 2 2 2 8 eta (q ) eta (q ) eta(i q) = ------------------------------------ 5 i pi ------ 48 e with its frizzy region of validity, http://gosper.org/cabbage.png , http://gosper.org/cabbage2.png.) Then we can go up and down by octaves via the easiest ("1,2,4", good for all n) eta relation eta(%e^-(2*%pi*sqrt(n)))^24 = eta(%e^-(4*%pi*sqrt(n)))^8*eta(%e^-(%pi*sqrt(n)))^16+16*eta(%e^-(4*% pi*sqrt(n)))^16*eta(%e^-(%pi*sqrt(n)))^8 24 1 8 1 16 1 eta (-------------) = eta (-------------) eta (-----------) 2 sqrt(n) pi 4 sqrt(n) pi sqrt(n) pi e e e 16 1 8 1 + 16 eta (-------------) eta (-----------) 4 sqrt(n) pi sqrt(n) pi e e (Jacobi's "aequatio identica satis abstrusa"). (If he only knew.) E.g., the double "angle": eta(%e^-(2*%pi*sqrt(n))) = ((prod(gamma(k/n)^jacobi(k,n),k,1,n-1))^2*eta(%e^-(%pi*sqrt(n)))^8*('prod(gamma(k/n)^jacobi(k,n),k,1,n-1))^2-4*%pi^2*eta(%e^-(%pi*sqrt(n)))^8*n^2)/(256*%pi^4*n^4))^(1/24) n - 1 /===\ 1 | | jacobi(k, n) k 2 8 1 eta(-------------) = expt(( | | gamma (-)) eta (-----------) 2 sqrt(n) pi | | n sqrt(n) pi e k = 1 e n - 1 /===\ | | jacobi(k, n) k 2 8 1 2 2 4 4 1 (( | | gamma (-)) - 4 eta (-----------) n pi )/(256 n pi ), --) | | n sqrt(n) pi 24 k = 1 (but n must be a Beegner.) --rwg