On 2015-10-21 06:21, Joerg Arndt wrote:
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) )
That's the log derivative.
E(q) = exp( - sum(n>=1, 1/n * q^n / (1-q^n) ) )
Yeah, I think Mma uses that in its QPochhammer numerics. In my bunch of special values of eta'/eta, I see a cryptic claim that 1/(8 π x DedekindEta[I x]^4) + (I Derivative[1][DedekindEta][I x])/(2 π DedekindEta[I x]^5) is algebraic whenever x is a positive quadratic surd. (Periodic or terminating CF.) E.g., Inadequate precision: In[49]:= RootApproximant[( DedekindEta[I x]/(4 x) + I Derivative[1][DedekindEta][I x])/( 2 \[Pi] DedekindEta[I x]^5) /. x -> GoldenRatio] Out[49]= Root[ 3 + 34 #1 + 15 #1^2 - 30 #1^3 + 10 #1^4 + 6 #1^5 - 5 #1^6 + 23 #1^7 &, 1] Adequate precision: In[50]:= RootApproximant[( DedekindEta[I x]/(4 x) + I Derivative[1][DedekindEta][I x])/( 2`22 \[Pi] DedekindEta[I x]^5) /. x -> GoldenRatio] Out[50]= Root[-6 - 62 #1 + 29 #1^2 - 3 #1^3 - 35 #1^4 + 27 #1^5 + 51 #1^6 &, 1] Superadequate precision: In[47]:= RootApproximant[( DedekindEta[I x]/(4 x) + I Derivative[1][DedekindEta][I x])/( 2`69 \[Pi] DedekindEta[I x]^5) /. x -> GoldenRatio] Out[47]= Root[-6 - 62 #1 + 29 #1^2 - 3 #1^3 - 35 #1^4 + 27 #1^5 + 51 #1^6 &, 1] In[51]:= RootApproximant[( DedekindEta[I x]/(4 x) + I Derivative[1][DedekindEta][I x])/( 2`22 \[Pi] DedekindEta[I x]^5) /. x -> GoldenRatio - 1/2] Out[51]= Root[-7 - 416 #1 + 63 #1^2 + 15 #1^3 + 181 #1^4 &, 1] In[52]:= RootApproximant[( DedekindEta[I x]/(4 x) + I Derivative[1][DedekindEta][I x])/( 2`33 \[Pi] DedekindEta[I x]^5) /. x -> GoldenRatio - 1/2] Out[52]= Root[-7 - 416 #1 + 63 #1^2 + 15 #1^3 + 181 #1^4 &, 1] In[55]:= RootApproximant[( DedekindEta[I x]/(4 x) + I Derivative[1][DedekindEta][I x])/( 2`33 \[Pi] DedekindEta[I x]^5) /. x -> (3 + Sqrt[17])/2] Out[55]= Root[-36 - 629 #1 - 225 #1^2 + 195 #1^3 &, 1] In[56]:= RootApproximant[( DedekindEta[I x]/(4 x) + I Derivative[1][DedekindEta][I x])/( 2`22 \[Pi] DedekindEta[I x]^5) /. x -> (3 + Sqrt[17])/2] Out[56]= Root[-36 - 629 #1 - 225 #1^2 + 195 #1^3 &, 1] Anybody recognize it? Maybe it can be tweaked to work for x a quadratic surd in the upper halfplane. --rwg
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). ) _______________________________________________