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 Tetra-series Gottfried Ultimate Fellow Posts: 766 Threads: 119 Joined: Aug 2007 11/23/2007, 10:47 AM (This post was last modified: 11/23/2007, 11:23 AM by Gottfried.) I'm extending a note, which I posted to sci.math.research today. I hope to find a solution of the problem by a reconsideration of the structure of the matrix of stirlingnumbers 1'st kind, which may contain "Infinitesimals" which become significant if infinite summing of its powers are assumed. ------------------------------- (text is a bit edited) ----------------- Due to a counterexample by Prof. Edgar (see sci.math) I have to withdraw this conjecture. The error may essentially be due to a misconception about the matrix of Stirling-numbers 1'st kind and the infinite series of its powers. ---- It is perhaps similar to the problem of the infinite series of powers of the pascal-matrix P, which could be cured by assuming a non-neglectable infinitesimal in the first upper subdiagonal Assuming for an entry $p_{r,r+1}$ of the first upper subdiagonal in row r (r beginning at zero) in the pascal matrix P $\hspace{24} p_{r,r+1} = binomial(r,r+1)$ which is an infinitesimal quantity and appears as zero in all usual applications of P. But if applied in a operation including infinite series of consecutive powers, then in the matrix Z, containing this sum of all consecutive powers of P we get the entries $\hspace{24} z_{r,r+1} = binomial(r,r+1)* \sum_{k=1}^{\infty} \frac1k$ By defining $\hspace{24} binomial(r,r+1) = \frac{r!}{(r+1)!(-1)!} = \frac1{(r+1)*(-1)!}$ assuming $\frac{\zeta(1)}{(-1)!} = - 1$ this leads to the non-neglectable rational quantities in that subdiagonal of the sum-matrix $\hspace{24} \begin{eqnarray} z_{r,r+1}& = &\frac1{(r+1)* (-1)!} * \sum_{k=1}^{\infty} \frac1k \\ & = &\frac1{(r+1)} * \frac{\zeta(1)}{(-1)!} \\ & = & - \frac1{(r+1)} \end{eqnarray}$ With that correction the infinite series of powers of the pascal-matrix leads then to a correct matrix, $\hspace{24} z_{r,c} = binomial(r,c)* \zeta(c-r)$ including the above definitions for the first upper subdiagonal, which provides the coefficients of the (integrals of) the bernoulli-polynomials, and can be used to express sums of like powers as expected and described by H.Faulhaber and J.Bernoulli. (for more details see powerseries of P, page 13 ff ) --- This suggests then to reconsider the matrix of Stirling-numbers of 1'st kind with the focus of existence of a similar structure in there. ------------------------------------------------------- Does this sound reasonable? It would require a description of the matrix of Stirling-numbers 1'st kind, which allows such an infinitesimal quantity. But there is one important remark: this matrix contains the coefficients of the series for logarithm and powers of logarithms. A modification of this matrix would then introduce an additional term in the definition of these series. Something hazardeous... 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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