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 Tetra-series Gottfried Ultimate Fellow Posts: 766 Threads: 119 Joined: Aug 2007 11/21/2007, 09:41 AM (This post was last modified: 11/21/2007, 10:45 AM by Gottfried.) andydude Wrote:Gottfried Wrote:I get for the sum of both by my matrix-method Code:```.   Sb(x) + Rb(x) = V(x)~ *Mb[,1] + V(x)~ * Lb[,1]                 = V(x)~ * (Mb + Lb)[,1]                 = V(x)~ *    I [,1]                 = V(x)~ *   [0,1,0,0,...]~                   = x   Sb(x) + Rb(x) = x``` This is what I have the most trouble understanding. First what is your [,1] notation mean? I understand "~" is transpose, and that Bb is the Bell matrix $Bb = B_x[s^x]$. Second, what I can't see, or is not obvious to me at least, is why: $(I + Bb^{-1})^{-1} + (I + Bb)^{-1} = I$ Is there any reason why this should be so? Can this be proven? Wait, I just implemented it in Mathematica, and you're right! (as right as can be without a complete proof). Cool! This may just be the single most bizarre theorem in the theory of tetration and/or divergent series. Andrew Robbins Hi Andrew - first: I appreciate your excitement! Yepp! :-) second: (The notation B[,1] refers to the second column of a matrix B) Yes, I just posed the question, whether (I+B)^-1 + (I+B^-1)^-1 = I in the sci.math- newsgroup. But the proof for finite dimension is simple. You need only factor out B or B^-1 in one of the expressions. Say C = B^-1 for brevity Code:```.    (I + B)^-1 + (I + C)^-1   = (I + B)^-1 + (CB + C)^-1 = (I + B)^-1 + (C(B + I))^-1 = (I + B)^-1 + (B + I)^-1*C^-1 = (I + B)^-1 + (B + I)^-1*B = (I + B)^-1 *(I + B) = I``` As long as we deal with truncations of the infinite B and these are well conditioned we can see this identity in Pari or Mathematica with good approximation. However, B^-1 in the infinite case is usually not defined, since it implies the inversion of the vandermonde matrix, which is not possible. On the other hand, for infinite lower *triangular* matrices a reciprocal is defined. The good news is now, that B can be factored into two triangular matrices, like B = S2 * P~ where P is the pascal-matrix, S2 contains the stirling-numbers of 2'nd kind, similarity-scaled by factorials S2 = dF^-1 * Stirling2 * dF (dF is the diagonal of factorials diag(0!,1!,2!,...) ) Then, formally, B^-1 can be written B^-1 = P~^-1 *S2^-1 = P~^-1 * S1 (where S1 contains the stirling-numbers of 1'st kind, analoguously factorial rescaled, and S1 = S2^-1 even in the infinite case) B^-1 cannot be computed explicitely due to divergent sums for all entries (rows of P~^-1 by columns of S1), and thus is not defined. However, in the above formulae for finite matrices we may rewrite C in terms of its factors P and S1, and deal with that decomposition-factors only and arrive at the desired result (I've not done this yet, pure lazyness...) third: This suggests immediately new proofs for some subjects I've already dealt with, namely all functions, which are expressed by matrix-operators and infinite series of these matrix-operators. For instance, I derived the ETA-matrix (containing the values for the alternating zeta-function at negative exponents) from the matrix-expression Code:```. ETA = (P^0 - P^1 + P^2 ....)      = (I + P)^-1```If I add the similar expression for the inverses of P I arrive at a new proof for the fact, that each eta(2k) must equal 0 for k>0. Yes- this is a very beautiful and far-reaching fact, I think ... Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Tetra-series - by Gottfried - 11/20/2007, 12:47 PM RE: Tetra-series - by andydude - 11/21/2007, 07:14 AM RE: Tetra-series - by Gottfried - 11/22/2007, 07:04 AM RE: Tetra-series - by andydude - 11/21/2007, 07:51 AM RE: Tetra-series - by Gottfried - 11/21/2007, 09:41 AM RE: Tetra-series - by Ivars - 11/21/2007, 03:58 PM RE: Tetra-series - by Gottfried - 11/21/2007, 04:37 PM RE: Tetra-series - by Gottfried - 11/21/2007, 06:59 PM RE: Tetra-series - by andydude - 11/21/2007, 07:24 PM RE: Tetra-series - by Gottfried - 11/21/2007, 07:49 PM RE: Tetra-series - by andydude - 11/21/2007, 08:39 PM RE: Tetra-series - by Gottfried - 11/23/2007, 10:47 AM RE: Tetra-series - by Gottfried - 12/26/2007, 07:39 PM RE: Tetra-series - by Gottfried - 02/18/2008, 07:19 PM RE: Tetra-series - by Gottfried - 06/13/2008, 07:15 AM RE: Tetra-series - by Gottfried - 06/22/2008, 05:25 PM Tetra-series / Inverse - by Gottfried - 06/29/2008, 09:41 PM RE: Tetra-series / Inverse - by Gottfried - 06/30/2008, 12:11 PM RE: Tetra-series / Inverse - by Gottfried - 07/02/2008, 11:01 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 10:38 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 11:01 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 01:25 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 02:40 PM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 09:37 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 10:33 PM RE: Tetra-series / Inverse - by Gottfried - 11/01/2009, 07:45 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 03:56 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 04:12 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 05:04 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 12:58 PM

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