Iteration series: Series of powertowers  "T geometric series"  Printable Version + Tetration Forum (https://math.eretrandre.org/tetrationforum) + Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) + Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) + Thread: Iteration series: Series of powertowers  "T geometric series" (/showthread.php?tid=38) Pages:
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Iteration series: Series of powertowers  "T geometric series"  Gottfried  08/24/2007 Hi  one remark at the beginning. I don't have another name than "powertower" for a singe tetrationterm, so I still use this word here. As you rememeber, at my initial interest were not the deep hardcore properties of the tetrationfunction, but rather some series, evaluated in my more general compilation of relations of important numbertheoratical coefficientsmatrices like binomials, stirlingnumbers, bernoullinumbers and the like (where tetration popped up as an iterative application) I guess, it is of worth, to add to the knowledgebase also that pieces about series of tetrationterms, aka powertowers. Here are two conjectures about identies of series; they are analogons to the known geometric series (if the tetrationexponent/iteration is 1 then the series are equal to geometric series). The first conjecture was already stated in different newsgroups. They have a strange, but fascinating charme... Gottfried [update]: I adapted the title to improve the directory of the threadslist RE: Series of powertowers  "T geometric series"  bo198214  08/24/2007 Hey Gottfried, that sounds really interesting, however give me some time to thoroughly read it. RE: Series of powertowers  "T geometric series"  Gottfried  08/25/2007 bo198214 Wrote:Hey Gottfried, Nice :) The requested time is generously granted... Btw: I've uploaded a slightly enhanced version of the article, where I smoothed the notation, added a short proof and added an explicite remark about the use of the vectors V(x) to prevent misreadings (which occured recently). Gottfried RE: Series of powertowers  "T geometric series"  andydude  09/06/2007 @Gottfried I read your paper, and your first theorem is interesting! Although I couldn't quite follow the "proof", it seems plausible, and I might recommend using an umbralcalculus style of proof. From what I understand umbral calculus is making transitions that are not rigorously true, but strictly syntactic substitutions. For example: which seems similar to a substitution you make in your paper. I think this would make the proof much clearer, and easier to follow, although I think it has gone out of style, and may be viewed as less rigorous than other methods of proof. I don't know if it helps or not, I just thought I'd comment on it. Andrew Robbins RE: Series of powertowers  "T geometric series"  Gottfried  09/06/2007 andydude Wrote:@Gottfried Hi Andrew  thanks for the comment! I know a bit about umbralcalculus from the context of Bernoullinumbers, where it is often used to prove some identities. However, even with your suggestion I do not know how to apply it here to improve readability (and reliablity) of the proof. But I'll give it a try. I would like to know, what I am missing. For me it seems easy, but, after a talk with a professor at the mathdepartment here at my university I am a bit discouraged concerning my abilities doing formal proofs...;) I missed all the classes of formal math education, when I moved from studying computerscience to socialassistance, so I'm doing numbertheory only as a hobby in spare time. Back to the proof. All what I employ is linear combinations and interchanging of order of summation, plus, and this may be the crucial point, to assume, that I can use the values of the etafunction as replacements of the infinite sums of cofactors. A notation is nothing more than a shortcut for the explicite, well known exponentialseries for all consecutive powers of y; say in a column c of the result vector (1) where from the elementary properties of the exponentialseries What I am doing then is to apply the linear combination of consecutive x, beginning at x=0 to that formula, expecting, that the result is again the corresponding linear combination: and where I expect, that this does not need a special proof (but may be, I'm in error already here) The crucial point is then to assume, that this is valid for infinite alternating series of (x_0^r  x_1^r + x_2^r  + ... ) , where x_k are the natural numbers, such that and that is interchangable for the linear combination of x_k. What we have is then the doublesum for a column c of the resultvector which is, after interchanging the order of summation (2) The lhs is now the sum but the interesting result is only in the column where c=1 so Then, on the rhs in (2), I use the fact, that each second eta0(r) = 0, and also I add the remaining eta0((2r+1)) with positive and negative signs to zero, which gives then the result (I omit here the other details in my article). Since powers of Bs are independent of the parameter of x and we can write the serial notation for each power of Bs in a similar form of (1), the reasoning down to (2) is exactly the same for any height of the towers. The only two possible problems, which I can see here, are the questions, whether the order of summation can be exchanged, and whether the linear combination of V(1)V(2)+V(3)V(4)... can be replaced by the etavalues of appropriate exponents. For the baseparameter s in the range 1/e^e < s < e^(1/e) the series are not too much diverging even for other heights of the towers, and since the sign is alternating, they can be regularly Eulersummed. This all is nothing else than to rewrite in serialnotation, what is implicite when using the notation of matrixmultiplication. Hmmm.... If there is something else missing, I would like to learn, what this is (perhaps I could even satisfy my partner of the short discussion here in the mathdepartement :)) Regards  Gottfried RE: Series of powertowers  "T geometric series"  hannibaal  01/31/2012 Hi hello I'm working now for same problem but, isn't in the sense of how developing Taylor series, we use Functional equation and identities in the place of summing notation as : S(i=1 to n)Xi=X1+X2+... more general use the functional composition of function (complex,; of two(and more) variables;f(x,y); but the signe of complex composion is fi(i=1 to n)(Bi,Ai*x)=f(B1,A1*f(B2,A2*f(...,f(Bn,An*x)...))) but this is a basic not all??, alos we need special function of functional function as Gfunction Barne, as Kn(z+1)=z^z^n*Kn(z) New i'm going of developed( i was developed without complex, so now i'm learn most of complex thoery), so much I beleive that mahematic must going beyond that traditional sumation and product (infinit), and etablish another infinit sense of series. thanks for you all. RE: Series of powertowers  "T geometric series"  Kouznetsov  02/03/2012 (08/24/2007, 02:44 PM)Gottfried Wrote: ... Here are two conjectures about identies of series;.. Gottfried1. Gottfried: Your "here" leads to http://go.helmsnet.de/math/pdf/Tetration_GS_short.pdf%20 causing diagnistics Not Found The requested URL /math/pdf/Tetration_GS_short.pdf was not found on this server. Could you check it please? 2. Luann Cole urgently needs to find bo198214 If anybody see him, let Bo communicate Luann, please. RE: Series of powertowers  "T geometric series"  Gottfried  02/03/2012 (02/03/2012, 03:06 PM)Kouznetsov Wrote:There is a trailing blank in your link. To access the file please just remove that trailing link in the addressfield of your browser. I'll also repair that link in my indexpage. Thanks for the notification!(08/24/2007, 02:44 PM)Gottfried Wrote: ... Here are two conjectures about identies of series;.. Gottfried1. Gottfried: Your "here" leads to The link is http://go.helmsnet.de/math/pdf/Tetration_GS_short.pdf Also, perhaps I should update that old file with my new knowledge: a) The conjectures for series of powertowers of height greater than 1 were shown to be not true by some correspondent. However, that concerns only one half of the doubly infinite series: either the series with indexes k = 0 to infinity or that with k=0 to + infinity is wrong using the matrixmethod and the other is correct. Moreover, my suspection is that the error is systematic and possibly can be compensated by some term similar to the procedure which we know from the Ramanujansummation, where we need to consider one additional integralexpression for the validity of the divergent Ramanujansummation. But I did not find yet an appropriate expression for this in the case of the iterationseries of powertowers/tetration. b) Some things could now be expressed less exploratory and hypothetical and instead more formal and firm. I'll see what i can do at the weekend... Gottfried RE: Series of powertowers  "T geometric series"  Kouznetsov  02/03/2012 (02/03/2012, 04:47 PM)Gottfried Wrote: The link is http://go.helmsnet.de/math/pdf/Tetration_GS_short.pdfThank you! Now it works. (02/03/2012, 04:47 PM)Gottfried Wrote: ..But I did not find yet an appropriate expression for this in the case of the iterationseries of powertowers/tetration. ..? Does your method allow to evaluate tetration faster (or more precise) than my one? There are already two wikis about tetration, http://tori.ils.uec.ac.jp/TORI/index.php/Tetration http://math.eretrandre.org/hyperops_wiki I tried to create an account at the second one and failed. I see you are successful.. Did you do it by yourself or Henryk had created your account? I should greatly appreciate your comments about http://tori.ils.uec.ac.jp/TORI RE: Series of powertowers  "T geometric series"  Gottfried  02/03/2012 Hello Dmitri  (02/03/2012, 06:19 PM)Kouznetsov Wrote:My conjecture, based on standard matrixidentities which I assumed could be extended to the case of infinite size, was the following. I constructed the doublyinfinite series of powertowers of increasing/decreasing height, so for the index/height h of infinity to +infinity. By the matrixidentities (involving the vonNeumannseries of the according Carlemanmatrix and its inverse) I expected that the sum of that doublyinfinite series was always zero.(02/03/2012, 04:47 PM)Gottfried Wrote: ..But I did not find yet an appropriate expression for this in the case of the iterationseries of powertowers/tetration. ..? But that was not true  but the difference to the expected value of zero was systematically distorted which shows a sinusoidal curve. I expect, that that curve has some sinusoidal function and that this function might be remotely related to that integral which we need if we do Ramanujansummation. I have described this more precisely at http://go.helmsnet.de/math/tetdocs/Tetraseriesproblem.pdf Some more introductory remarks to the concept of iterationseries are in the introductory remarks at my tetrationhomepage at http://go.helmsnet.de/math/tetdocs Quote:Does your method allow to evaluate tetration faster (or more precise) than my one?No, this is just the (re)discovery of the concept of "Carlemanmatrix" which I did not know when I came across the problem of tetration and my general matrixapproach to some numbertheoretic problems. So it has the known deficients of the Carlemanmatrixapproach: a) there is nothing known yet which would make the Carlemanmatrixapproach a unique preferable solution, b) the solution for the fractional iterates is dependent on the fixpoint which was chosen to center the power series around, and c) we get complexvalued power series for real valued fractional heights. Because of that unsolved problems I've put my studies on low energy and I hoped, that your solution would come out as *the* general (and generally accepted) method for the tetration. I'd really like to see that this would happen! Quote:There are already two wikis about tetration,Yes I know them; however their ambition seem to be very high and I have only undergraduate courses in mathematics in computerscience in the 70ties, and my numbertheoryknowledge is amateurishly compiled singletopicmaterial. So I felt I could not contribute on that formal/rigorous level of definition and knowledge. But perhaps there is something I can help with... Quote:I tried to create an account at the second one and failed. I see you are successful.. Did you do it by yourself or Henryk had created your account?Well, that was Henryk  now that you call for him I'm getting aware that we've heard little or nothing of him lastly... It's very nice to hear from you now! Gottfried 