Maybe the following is helpful (eta-Lambert identities). E(q) := prod(n>=1, 1 - q^n ) and the derivative is with respect to q. q * E(q)' / E(q) = - sum(n>=1, n * q^n/(1-q^n) ) E(q) = exp( - sum(n>=1, 1/n * q^n / (1-q^n) ) ) The first is straight forward (product rule), for the second see page 452 (exercise 8 for chapter XII) in Konrad Knopp, Theory And Application Of Infinite Series, (1954) http://www.archive.org/details/theoryandapplica031692mbp Best regards, jj * Bill Gosper <billgosper@gmail.com> [Oct 21. 2015 14:00]:
but I sure missed it. Pi 1 4 2 E (-)! 1 + 2 n 4 Sum[------------------------------, {n, 0, oo}] == ----------- 2 n Pi -2 (1 + 2 n) Pi 3 E (1 - E ) Pi
I got this as the q -> E^(-2 Pi) special case of Sum[DivisorSigma[1,2*n+1] q^n,{n,0,∞}]== Sum[(1 + 2 n) q^n/(1 - q^(1 + 2 n)), {n, 0, ∞}] == ( I (Derivative[1][DedekindEta][-((I Log[q])/(4 \[Pi]))]/ DedekindEta[-((I Log[q])/(4 \[Pi]))] - ( 3 Derivative[1][DedekindEta][-((I Log[q])/(2 \[Pi]))])/ DedekindEta[-((I Log[q])/(2 \[Pi]))] + ( 2 Derivative[1][DedekindEta][-((I Log[q])/\[Pi])])/ DedekindEta[-((I Log[q])/\[Pi])]))/(2 \[Pi] Sqrt[q])
As suggested by Sunday's "notational modularity" item, I have a bunch more special values of eta and eta' to crank into this. (I bet you can hardly wait.) Also, eta[q], eta[q^2], and eta[q^4] are polynomially related by Jacobi's aequatio identica satis abstrusa <https://www.google.com/search?client=safari&rls=en&q=aequatio+identica+satis+abstrusa&ie=UTF-8&oe=UTF-8> , which should have an eta' analog. --rwg Vaguely related: ArcLength[{t, Sqrt[Cos[t]]}, {t, -π/2, π/2}] == Integrate[Sqrt[1 + (1/4)*Sin[t]*Tan[t]], {t, -Pi/2, Pi/2}] but the ISC says HUH? Really? (Connection: Area under that arc = π^(3/2)/(2 Sqrt[2] (1/4)!^2). ) _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun