rwg>>Still open: eta'', eta(e^-(pi phi)), e.g.. rwg> Well, not the latter, which is a singular value. NO! i phi is *not* a modular transformation of i sqrt 5. I have no idea what eta(e^-(pi phi)) is. But here is some phi and sqrt 5 merriment: 'sum(n/(%e^(2*sqrt(5)*%pi*n)-1),n,1,inf) = -%phi*Gamma(1/20)^2*Gamma(9/20)^2/(960*5^(3/4)*%pi^3)-1/(8*sqrt(5)*%pi)+1/24 inf 2 1 2 9 ==== %phi Gamma (--) Gamma (--) \ n 20 20 > --------------------- = - -------------------------- / 2 sqrt(5) %pi n 3/4 3 ==== %e - 1 960 5 %pi n = 1 1 1 - ------------- + -- 8 sqrt(5) %pi 24 'sum(n/(%e^(2*%pi*n/sqrt(5))-1),n,1,inf) = %phi*Gamma(1/20)^2*Gamma(9/20)^2/(192*5^(3/4)*%pi^3)-sqrt(5)/(8*%pi)+1/24 inf 2 1 2 9 ==== %phi Gamma (--) Gamma (--) \ n 20 20 sqrt(5) 1 > ------------- = -------------------------- - ------- + -- / 2 %pi n 3/4 3 8 %pi 24 ==== ------- 192 5 %pi n = 1 sqrt(5) %e - 1 Joerg Arndt> [arxiv table of rational Gamma reductions] rwg>That's the sort of thing I'd much prefer be done by an active CAS. Same goes for
most of http://functions.wolfram.com .
Notice that the above results are considerably simpler than the 5th singular values tabulated in MathWorld and Borwein&Borwein (p 298), due partly to using the 20th singular value instead, but more to a Gamma simplification that was not in http://arxiv.org/abs/math/0403510 . If we can't find a rigorous algorithm to simplify them, there's always LatticeReduce! That's how I'm finding the (eta')s. --rwg PS, these log derivate sums can also be written 'sum(n/(1/q^n-1),n,1,inf) = 'sum(q^n/(1-q^n)^2,n,1,inf) inf inf ==== ==== n \ n \ q > ------- = > --------- / -n / n 2 ==== q - 1 ==== (1 - q ) n = 1 n = 1