I tried to give this a shot but got hung up by requests by maxima for additional assumptions; it wants to know the sign of the variable a, but from skimming that paper it looks like we don't want to assume a particular sign for a,b, or c. I guess it might be possible to exhaustively do all sign cases but maybe there is a better way.
-M. Hampton On Feb 20, 1:22 am, Alexander Povolotsky <[email protected]> wrote: > Hi, > > S. K. Lukas in > > http://www.math.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf<http://www.math.jmu.edu/%7Elucassk/Papers/more%20on%20pi.pdf> > > derived several identities, which relate Pi (via definite integrals) with > the several few Pi fractional convergents, which denominators and numerators > are described in > > http://www.research.att.com/~njas/sequences/A002486<http://www.research.att.com/%7Enjas/sequences/A0024865>http://www.research.att.com/~njas/sequences/A002485<http://www.research.att.com/%7Enjas/sequences/A0024865> > > I raised the issue re the possibility of deriving generalized ("n" parameter > based ) definite integral identity relating Pi with ALL (each at its own > value of n) Pi fractional convergents > (referenced in above sequences) - see my exchange with S. K. Lukas. > > Unfortunately I do not have sufficient computational resources (I do > not have access to Maple or Mathematica, instead I have Pari/GP installed > on > my very old home computer) to take advantage of Stephen's generous > offer to play with his Maple program, which he wrote and which is listed > in > > > http://www.math.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf<http://www.math.jmu.edu/%7Elucassk/Papers/more%20on%20pi.pdf> > > Quoting William Stein's kind reply to me: > "The program is on page 9 of the linked to pdf and > that it is literally only 18 lines of code, and would likely be very > easy for somebody who knows *both* Sage and Maple to port to Sage." > > May one of you could help me by modifying (porting to SAGE) and running > Stephen's > program towards experimental attempt of deriving desired generalized > identity? > > Thanks, > Best Regards, > Alexander R. Povolotsky > > > ------------------------------------------------- > > On Tue, Feb 10, 2009 at 8:03 PM, Stephen Lucas <[email protected]> wrote: > >> Alexander, > >> Thanks for your email, I could only get to it now due to a heavy day > >> of teaching. > > >> I am tempted to say that what you have found are essentially > >> coincidences. > > >> However, don't let me dissuade you from searching for patterns -- I > >> may indeed have missed them. If you want to experiment with my Maple > >> code, go right ahead. > > >> Incidentally, since you read the paper from my website, you may be > >> interested to know that an edited version of it (slightly condensed > >> without the Maple code) was published this month (February 2009) in > >> the American Mathematical Monthly. > > >> Yours sincerely, > >> Steve Lucas > > >> On Mon, Feb 9, 2009 at 8:00 PM, Alexander Povolotsky <[email protected]> > wrote: > >>> Hello Dear Stephen, > > >>> I found and read your article "Integral approximations to Pi with > >>> nonnegative integrands" > >>>http://www.math.jmu.edu/~lucassk/Papers/more%20on%20pi.pdf<http://www.math.jmu.edu/%7Elucassk/Papers/more%20on%20pi.pdf> > > >>> I looked specifically into the "make-up" of your formula > > >>> Pi = 355/113-1/3164*Int(x^8*(1-x)^8*(25+816*x^2)/(1+x^2),x = 0 .. 1) > > >>> I noticed that > >>> 3164 / (816 - 25) = 3164/791= 3164 / (7 * 113) = 4 > >>> where originally (incorrectly) I thought that 7 is den of 22/7 and > >>> 113 is den of 355/113 ... > >>> also we observe that 113 - 7 = 106 > > >>> Then I looked at your another formula > > >>> Pi = 104348/33215 -1/38544*Int(x^12*(1-x)^12*(1349 > >>> -1060*x^2)/(1+x^2),x = 0 .. 1) > > >>> I noticed that > >>> 38544 / (1349 + 1060) = 38544 / 2409 = 38544 / (3 * 11 * 73 ) = 16 > >>> is it interesting that 33 + 73 = 106 as above ? > > >>> Also one could see that in your formula for 103993/33102 > > >>> 755216 / (124360 - 77159) = 755216 / 47201 = 755216 / (7 * 11 * 613) = > 16 > > >>> as well as that in your formula for 333/106 > > >>> 530 / (462 - 197) = 530 / 265 = 530 / (5 * 53) = 2 > > >>> Is it some sort of pattern there with regards to ratios ? > > >>> Actually the construct to get the ratios is the same - (if i am not > >>> mistaken) it takes in account the initial sign of the involved terms. > > >>> Thanks, > >>> Regards, > >>> Alexander R. Povolotsky > >> -- > >> Stephen Lucas, Associate Professor > >> Department of Mathematics and Statistics > >> MSC 1911, James Madison University, Harrisonburg, VA 22807 USA > >> Phone 540 568 5104, Fax 540 568 6857, Web > > http://www.math.jmu.edu/~lucassk/<http://www.math.jmu.edu/%7Elucassk/> > > >> Email lucassk at jmu dot edu (Work) stephen.k.lucas at gmail dot com --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
