Dear Dr. Essomba, Thank you for catching my D111 mistake. Your claim is immediately substantiated by the limit 𝜃→0, which gives 1 = ¼. And confirmed by rederivation, simply multiplying D110 by itself with 𝜃→√3𝜃. There are several errors on that page (which needs to be done over in Mathematica) due to faulty conversion to html, (e.g., p for π, f for 𝜙,...) but this isn't one of them, since the error is repeated in C111. Puzzling. Also, here is experimental confirmation of your 2017-10-13 07:51 recurrence: In[502]:= NestList[(Print[√√(2^(4 #[[1]] + 5) (3 - 2 √(2 + 2 #[[2]]) + #[[2]]))]; {#[[1]] + 1, √(2 + 2 #[[2]])/2}) &, {0, -1.}, 9] During evaluation of In[502]:= 2.82842712474619 During evaluation of In[502]:= 3.061467458920717 During evaluation of In[502]:= 3.12144515225806 During evaluation of In[502]:= 3.136548490545999 During evaluation of In[502]:= 3.140331156951562 During evaluation of In[502]:= 3.141277250930965 During evaluation of In[502]:= 3.141513800680598 During evaluation of In[502]:= 3.141572952943204 During evaluation of In[502]:= 3.141587832038716 Out[502]= {{0, -1.}, {1, 0.}, {2, 0.7071067811865476}, {3, 0.9238795325112867}, {4, 0.9807852804032304}, {5, 0.9951847266721969}, {6, 0.9987954562051724}, {7, 0.9996988186962042}, {8, 0.9999247018391445}, {9, 0.9999811752826011}} where u_n (= #[[2]]) converges to 1 while a_n converges to π⁴. --rwg Date: 2017-10-16 05:35 From: François Mendzina Essomba via math-fun <math-fun@mailman.xmission.com> To: math-fun@mailman.xmission.com, m_essob@yahoo.fr Reply-To: François Mendzina Essomba <m_essob@yahoo.fr>, math-fun < math-fun@mailman.xmission.com> Hello, I found in the page http://www.tweedledum.com/rwg/idents.htm strange identities, as incredible as each other ... I have observed at length some of the identities ranging from C105 to C115. And this is what I deduced from it: (9) D106 product(2*cos(theta/3^(n/m))-1,n=1..infinity)=product(cos(3^(i/m)*theta/2),i=0...m-1) ; (10) D105 product(2*cos(theta/2^(n/m))-1,n=1..infinity)=product((1+2*cos(2^(i/m)*theta))/3,i=0..m-1); (11) D110 product((2*cos(theta/3^(n/m))+1)/3,n=1..infinity)=(2^m/(theta)^m)*product(sin(3^(i/m)*theta/2)/3^(i/m),i=0..m-1); However I am a little perplexed as to the last formula that differs from Gosper's D111 identity by a factor of 4. The result I get is this: (12) D111 product((2*cos(theta/3^(n/2))+1)/3,n=1..infinity)=(2^2/(theta)^2)*product(sin(3^(i/2)*theta/2)/3^(i/2),i=0..2-1); I suppose it must be the same for this small series, I have yet to check. Best regards