Joerg>A collection of (currently) 5416 such identities by Michael Somos is available at http://eta.math.georgetown.edu/ Enjoy! * Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> [Oct 01. 2010 09:35]:

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These identities seem to fail for complex values. E.g., for the simplest, good old aequatio satis abstrusa, In[318]:= Equal[0, 16*(DedekindEta[-((Log[q]*I)/(2*Pi))])^8*(DedekindEta[-((Log[q^4]* I)/(2*Pi))])^16 + (DedekindEta[-((Log[q]*I)/(2* Pi))])^16*(DedekindEta[-((Log[q^4]*I)/(2* Pi))])^8 - (DedekindEta[-((Log[q^2]*I)/(2*Pi))])^24] In[319]:= %318 /. q -> Exp[-\[Pi]*(I + Sqrt[5])] Out[319]= False (*yet*) In[320]:= %318 /. q -> Exp[-\[Pi]*(+Sqrt[5])] Out[320]= 0 == -DedekindEta[I Sqrt[5]]^24 + DedekindEta[(I Sqrt[5])/2]^16 DedekindEta[2 I Sqrt[5]]^8 + 16 DedekindEta[(I Sqrt[5])/2]^8 DedekindEta[2 I Sqrt[5]]^16 Can they be restated to hold more broadly? --rwg BtW, DedekindEta[1/2 + (I Sqrt[5])/2] == ((-1)^(1/24) Sqrt[ Gamma[1/20] Gamma[9/20]])/(2 5^(5/16) GoldenRatio^(1/8) \[Pi]^(3/4))