[math-fun] formula for the catalan constant, eisenstein series reformulated and other findings.
hello, I just found this formula for Catalan constant , is this known ? /infinity \ /infinity \ | ----- | | ----- | | \ 1 | | \ 1 | 11 | ) -------------------| - 71/2 | ) ---------------------| | / 2 | | / 2 | | ----- n (cosh(Pi n) - 1)| | ----- n (cosh(2 Pi n) - 1)| \ n = 1 / \ n = 1 / /infinity \ | ----- | | \ 1 | + 11 | ) ---------------------| | / 2 | | ----- n (cosh(4 Pi n) - 1)| \ n = 1 / gives : 11*Sum(1/n^2/(cosh(Pi*n)-1),n = 1 .. infinity)-71/2*Sum(1/n^2/(cosh(2*Pi*n)-1), n = 1 .. infinity)+11*Sum(1/n^2/(cosh(4*Pi*n)-1),n = 1 .. infinity) when lprinted and evalf(%) gives : 0.91596559417721901505460351493238411077414937428167213426649811962176301977\ 625476947935651292611510624857442261919619957903589881... I think it is valid, one thing I know : I never saw it before. if I am not mistaking, that formula can be used to compute catalan constant to a high precision. I have others that are of interest here : http://pictor.math.uqam.ca/~plouffe/inspired3.pdf a html version is available also at http://pictor.math.uqam.ca/~plouffe/ see the middle of the page some of the text is in french but formulas are in universal language with <sigma> of course, this is a draft of course. some findings are related to Eisenstein series, that I find simpler to grasp in some way. ah yes, I have also found a generalization of Ramanujan formula for rational arguments, as you will see in the documents, instead of having only the cases with 4n+1 there is now a general formula for all odd index as you will see. bonne lecture, simon plouffe
Hello Simon, Pardon me if my question looks naïve, I see that you have listed formulae for 1 of the form p1/q1 sum( n^(4k+3) / (e^(pi n)-1) + p2/q2 sum( n^(4k+3) / (e^(2 pi n)-1) Do you think that one could one day find similar ones for exponent 5,9,13, ... (I am sure you already searched for them), since you got one for exponent 1 ? Olivier On Sun, Jun 14, 2009 at 18:44, Simon Plouffe<simon.plouffe@gmail.com> wrote:
hello,
I have others that are of interest here : http://pictor.math.uqam.ca/~plouffe/inspired3.pdf
Hello, about the '1' that appears in the formulas : there is a bunch of formulas for 0 too, that explains why I do not have it for 1. If you look closely to the formulas for 1, these formulas are the Eisenstein series reformulated, since the exponent 5, 9,13 where already known I looked for formulas for 3,7,11,15... actualy there are formulas for them BUT it uses 2 arguments instead of 1 : the coefficients are the SAME as the ones for eisenstein series, the trick is to see these formulas as particular Lambert series with the x replaced by exp(pi). Also, I find it really fascinating that I can find one formula for Pi and 1/Pi, Pi^2 and 1/Pi^2, by reverting the exponent for n. Catalan constant is <b(2)>, there are no formulas for b(4), b(6), etc. These are Dirichlet series, like b(4) = sum((-1)^(n+1)/(2*n+1)^4,n=1..infinity), no formulas for 1/b(4) and 1/b(6) either. one for Catalan but NONE for 1/Catalan, this is strange isn't ? I have tried with sinh, and all sorts of variants to find where was the bug about that : there are none, it seems that 1/Catalan cannot be found using these sums and I can't say or see why. simon plouffe
participants (2)
-
Olivier Gerard -
Simon Plouffe