Gosper's identities (Bill Gosper); Hello, Thank you dear Professor Bill Gosper, for the details related to my recurrence, I did not verify it experimentally. The negative terms (-2) ^ n or (-3) ^ n of this identities seem to me so rich and so profound ... It took me longer to understand Best regards. Le Samedi 21 octobre 2017 20h00, "math-fun-request@mailman.xmission.com" <math-fun-request@mailman.xmission.com> a écrit : Send math-fun mailing list submissions to     math-fun@mailman.xmission.com To subscribe or unsubscribe via the World Wide Web, visit     https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun or, via email, send a message with subject or body 'help' to     math-fun-request@mailman.xmission.com You can reach the person managing the list at     math-fun-owner@mailman.xmission.com When replying, please edit your Subject line so it is more specific than "Re: Contents of math-fun digest..." Today's Topics:   1. 31 lines of zero (mod 15) puzzle. (Ed Pegg Jr)   2.  Gosper's identities (Bill Gosper)   3. Gosper's nonidentities (Bill Gosper) ---------------------------------------------------------------------- Message: 1 Date: Fri, 20 Oct 2017 23:00:36 -0500 From: Ed Pegg Jr <ed@mathpuzzle.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] 31 lines of zero (mod 15) puzzle. Message-ID:     <CAGog0+Y90bRpf2mWS9eZ_Ho3DbAdt6UHVWvNfGCRev2CbLqm7g@mail.gmail.com> Content-Type: text/plain; charset="UTF-8" Here's an orchard of 31 lines of three on 15 points.  That might be maximal according to A003035. https://i.imgur.com/w18qs7R.jpg Arrange the fifteen numbers -7 to 7 in the disks so that the (mod 15) sum of all lines is zero. --Ed Pegg Jr ------------------------------ Message: 2 Date: Sat, 21 Oct 2017 02:22:35 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun]  Gosper's identities Message-ID:     <CAA-4O0E0cmKVasfuY65rGjf496FDtY8anutimpgecOeLyngZgw@mail.gmail.com> Content-Type: text/plain; charset="UTF-8" Dear Dr. Essomba, Thank you for catching my D111 mistake. Your claim is immediately substantiated by the limit ??0, which gives 1 = ?. And confirmed by rederivation, simply multiplying D110 by itself with ???3?. 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 ?, f for ?,...) 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  ??.  --rwg Date: 2017-10-16 05:35 From: Fran?ois Mendzina Essomba via math-fun <math-fun@mailman.xmission.com> To: math-fun@mailman.xmission.com, m_essob@yahoo.fr Reply-To: Fran?ois 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 ------------------------------ Message: 3 Date: Sat, 21 Oct 2017 10:15:19 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Gosper's nonidentities Message-ID:     <CAA-4O0ER2aMjEHX2+SdqNzGc0rpkfXApmnGrx4jX9mWAzuNT0g@mail.gmail.com> Content-Type: text/plain; charset="UTF-8" While collecting reasons why Mathematica's StirlingS1 and StirlingS2 should, like Macsyma, admit negative and fractional arguments, I found, to my discomfort,

From rwg@NEWTON.Macsyma.COM  Wed Jul 24 03:00:00 1996 Received: from optima.CS.Arizona.EDU by cheltenham.cs.arizona.edu; Wed, 24 Jul 1996 03:03:19 MST Received: from NEWTON.Macsyma.COM by optima.cs.arizona.edu (5.65c/15) via SMTP id AA07538; Wed, 24 Jul 1996 03:03:16 MST Received: from SWEATHOUSE.macsyma.com ([192.233.166.105]) by NEWTON.Macsyma.COM via INTERNET with SMTP id 363715; 24 Jul 1996 06:01:03-0400 Date: Wed, 24 Jul 1996 03:00-0700 From: Bill Gosper <rwg@NEWTON.macsyma.com> Reply-To: rwg@NEWTON.macsyma.com Subject: funny-looking sum To: math-fun@cs.arizona.edu Message-Id: <19960724100013.6.RWG@SWEATHOUSE.macsyma.com>

(Using Knuthian (nonnegative) Stirlings) inf    n - 2 ====  ====    x        k + 1 \      \    (%e  - x - 1)      stirling_s1(n, n - k)    - x n   >    ( >    ----------------------------------------) %e      = log(x + 1) /      /                    (k + 1)! ====  ==== n = 1  k = 0 apparently for x>=0, anyway.  (Convergence rate is useless.) ----------------------------------- It doesn't even slightly work, presumably mistranscribed.  To my active distress, Eric Weisstein has faithfully quoted this bogon as eqn (26) in http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html, (except my S is his |S|).  I am so far unable to guess its correct form, which presumably existed because I would have Taylor-expanded the ???? out of anything this weird.  Help?  --rwg ------------------------------ Subject: Digest Footer _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun ------------------------------ End of math-fun Digest, Vol 176, Issue 31 *****************************************