On Thu, Apr 3, 2014 at 4:33 PM, Bill Gosper <billgosper@gmail.com> wrote [...]
WDS> --These questions are just new questions...
If you were to take the attitude you are not even going to permit elliptic/modular functions to appear, then it might be the reflection and ntupling is all there is and Chowla & Selberg might be irrelevant.
rwg>A tool like "ToElliptic" would be valuable, and might synergize with this duction hack, but the user should keep control. If my memory is right about eta(e^-(pi sqrt(n/d))) always coming out in Gammas, there remains a chance of finding an "inaccessible" identity. --rwg
I finally see what Warren was driving at. My "ductions" list has, for (n/24)!, (5/24)! -> (5 Sqrt[1/2 (-1 + Sqrt[2]) (-1 + Sqrt[3])] (1/24)! (1/3)!)/(2 (1/6)!), (7/24)! -> (7 (1/24)! Sqrt[( 5 (-1 + Sqrt[3]) \[Pi] Sin[\[Pi]/24] Sin[(5 \[Pi])/24])/((1/12)! (5/ 12)!)])/(6 3^(1/4)), (11/24)! -> (11 Sqrt[5 (1 + Sqrt[3]) \[Pi]] (1/24)! (1/3)! Sin[\[Pi]/24])/( 4 2^(1/4) 3^(3/4) (1/6)! Sqrt[(1/12)! (5/12)!]), I.e., reduction to 1/24, but no further. But I find in my notes \[Eta][(1/(E^((Sqrt[2] * Pi)/(Sqrt[3]))))]== ((Gamma[1/24] * (Tan[Pi/24])^(1/4) * (Sin[Pi/8])^(1/6))/(2 * 2^(1/12) * 3^(1/8) * Sqrt[Gamma[1/12]] * Sqrt[Pi])) i.e., we can reduce to (1/12)! by allowing Dedekind eta. eta satisfies infinitely many relations like 0==27 * (\[Eta][q])^3 * (\[Eta][q^9])^9 + 9 * (\[Eta][q])^6 * (\[Eta][q^9])^6 + (\[Eta][q])^9 * (\[Eta][q^9])^3 - (\[Eta][q^3])^12] but as far as I can tell, the Gammas just cancel out. --rwg BtW, the more I understand http://en.wikipedia.org/wiki/Dedekind_eta_function , the more it sucks.