A generalization: SUM(COS(N*T)/SINH(N*X)^2,N,1,INF) = -THETADERIV[1](%PI/3,2,%E^-(X/3))/(12*SQRT(3)*ETA(%E^-(2*X)))+THETADERIV[1](T/2,2,%E^-X)/(4*THETA[1](T/2,%E^-X))+1/6 inf %pi - x/3 ==== thetaderiv (---, 2, %e ) \ cos(n t) 1 3
---------- = - ---------------------------- / 2 - 2 x ==== sinh (n x) 12 sqrt(3) eta(%e ) n = 1
t - x thetaderiv (-, 2, %e ) 1 2 1 + ------------------------ + - t - x 6 4 theta (-, %e ) 1 2 A related result: 'SUM(COS(N*T)*COSH(N*X)/SINH(N*X)^2,N,1,INF) = THETADERIV[4](T/2,2,%E^-X)/(4*THETA[4](T/2,%E^-X))-THETADERIV[1](%PI/3,2,%E^-(X/3))/(12*THETA[1](%PI/3,%E^-(X/3)))-1/12 inf t - x ==== thetaderiv (-, 2, %e ) \ cos(n t) cosh(n x) 4 2
------------------ = ------------------------ / 2 t - x ==== sinh (n x) 4 theta (-, %e ) n = 1 4 2
%pi - x/3 thetaderiv (---, 2, %e ) 1 3 1 - ---------------------------- - -- %pi - x/3 12 12 theta (---, %e ) 1 3 And an oddity: 'DIFF(ETA(Q),Q) = -THETADERIV[1](0,3,SQRT(Q))/(48*Q*ETA(Q)^2) thetaderiv (0, 3, sqrt(q)) d 1 -- (eta(q)) = - -------------------------- dq 2 48 q eta (q) That's theta[1]'''(0,sqrt q). --rwg On Sat, Apr 2, 2011 at 5:43 AM, Bill Gosper <billgosper@gmail.com> wrote:
I didn't realize we could do these. Sum[Csch[n*x]^2, {n, 1, Infinity}] == 1/6 + Derivative[0, 1, 0][EllipticThetaPrime][1, Pi/3, E^(-x/3)]/ (6*Sqrt[3]*DedekindEta[(I*x)/Pi])
Mma doesn't realize it either: In[517]:= % /. x -> 1/2
Out[517]= False Yet In[518]:= List @@ %% /. x -> 1/2
Out[518]= {4 (-1 + QPolyGamma[1, 1, Sqrt[E]] + QPolyGamma[1, 1 - 2 I Pi, Sqrt[E]]),
(0,1,0) Pi 1 EllipticThetaPrime [1, --, ----] 3 1/6 1 E - + --------------------------------------} 6 I 6 Sqrt[3] DedekindEta[----] 2 Pi
In[519]:= N[%, 33]
Out[519]= {4.74640293405957128234200051279399 + 0.*10^-35 I, 4.74640293405957128234200051279836} --rwg