inf 4 1 ==== Gamma (-) \ n 4 1 1
----------- = --------- - ----- + --. / %pi n 3 4 %pi 24 ==== %e - 1 64 %pi n = 1
Still open: eta'(e^-(pi sqrt r)),
'at('diff(eta(q),q,1),q = %e^-(2*sqrt(3)*%pi)) = %e^(2*sqrt(3)*%pi)*Gamma(1/3)^(3/2)*(2^(1/3)*sqrt(3)*gamma(1/3)^6+32*%pi^3)/(512*2^(1/3)*3^(3/8)*%pi^5) | d | -- (eta(q))| dq | - 2 sqrt(3) %pi |q = %e 2 sqrt(3) %pi 3/2 1 1/3 6 1 3 %e gamma (-) (2 sqrt(3) gamma (-) + 32 %pi ) 3 3 = -------------------------------------------------------------- 1/3 3/8 5 512 2 3 %pi 'at('diff(eta(q),q,1),q = %e^-(2*%pi/sqrt(3))) = 3*%e^(2*%pi/sqrt(3))*Gamma(1/3)^(3/2)*(32*%pi^3-2^(1/3)*sqrt(3)*Gamma(1/3)^6)/(512*2^(1/3)*3^(1/8)*%pi^5) | d | 2 %pi -- (eta(q))| - ------- dq | sqrt(3) |q = %e 2 %pi ------- sqrt(3) 3/2 1 3 1/3 6 1 3 %e Gamma (-) (32 %pi - 2 sqrt(3) Gamma (-)) 3 3 = ---------------------------------------------------------- 1/3 1/8 5 512 2 3 %pi
and thus the corresponding sums like the above.
'sum(n/(%e^(2*%pi*n/sqrt(3))-1),n,1,inf) = 3*2^(1/3)*Gamma(1/3)^6/(256*%pi^4)-3/(8*sqrt(3)*%pi)+1/24 inf 1/3 6 1 ==== 3 2 Gamma (-) \ n 3 3 1
------------- = ---------------- - ------------- + -- / 2 %pi n 4 8 sqrt(3) %pi 24 ==== ------- 256 %pi n = 1 sqrt(3) %e - 1
'sum(n/(%e^(2*sqrt(3)*%pi*n)-1),n,1,inf) = -2^(1/3)*Gamma(1/3)^6/(256*%pi^4)-1/(8*sqrt(3)*%pi)+1/24 inf 1/3 6 1 ==== 2 Gamma (-) \ n 3 1 1
--------------------- = - -------------- - ------------- + -- / 2 sqrt(3) %pi n 4 8 sqrt(3) %pi 24 ==== %e - 1 256 %pi n = 1
These were a lot of work, so they'd better be new! Still open: eta'', eta(e^-(pi phi)), e.g.. --rwg