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 Status of proofs Gottfried Ultimate Fellow Posts: 758 Threads: 117 Joined: Aug 2007 08/24/2007, 10:06 PM (This post was last modified: 08/27/2007, 11:00 AM by Gottfried.) [edit 25.8, minor textual corrections] Hmm, my thoughts may be amateurish, but I've time for a bit of discussion of it currently. That, what I called "my matrix method" is nothing else than a concise notation for the manipulations of exponential- series, which constitute the iterates of x, s^x, s^s^x . The integer-iterates. It comes out,that the coefficients for the series, which has to be summed to get the value for $\hspace{24} t_s^{(m)}\left(x\right) = s^{s^{\dots^s^x}} \text{with m-fold occurence of } s$ have a complicated structure, but which is anyway iterative with a simple scheme. It is a multisum, difficult to write, so I indicate it for the first three iterates m=1,m=2,m=3 m=1: $\hspace{24} T^{\tiny(1)}_s(x)= \sum_{k=0}^{\infty} x^k * \frac{log(s)^k}{k!}$ m=2: $\hspace{24} T^{\tiny(2)}_s(x)= \sum_{k_2=0}^{\infty} \sum_{k_1=0}^{\infty} x^{k_2} * k_1^{k_2} * \begin{pmatrix} k_1+k_2 \\ k_2 \end{pmatrix} * \frac{ log(s)^{ k_1+k_2}}{(k_1+k_2)!}$ m=3: $\hspace{24} T^{\tiny(3)}_s(x)= \sum_{k_3=0}^{\infty} \sum_{k_2=0}^{\infty} \sum_{k_1=0}^{\infty} x^{k_3} * k_2^{k_3} k_1^{k_2} * \begin{pmatrix} k_1+k_2 +k_3 \\ k_2,k_3 \end{pmatrix} * \frac{ log(s)^{ k_1+k_2+k_3}}{(k_1+k_2+k_3)!}$ In m>2 the binomial-term expands to a multinomial-term. When denoting $\hspace{24} n = k_1 + k_2 + ... + k_m$ then the notation for the general case m>1 can be shortened: m=arbitrary, integer>0: $\hspace{24} T^{\tiny(m)}_s(x)= \sum_{ k_m...k_1 =0..\infty \\ k_1+k_2+...+k_m=n } x^{k_m} * \left( k_{m-1}^{k_m} \dots k_2^{k_3} k_1^{k_2} \right) * \begin{pmatrix} n \\ k_2,k_3,...,k_m \end{pmatrix} * \frac{ log(s)^n}{n!}$ (hope I didn't make an index-error). This formula can also be found in wrong reference removed (update) Convergence: Because we know, that powertowers of finite height are computable by exponential-series we know also, that for the finite case this sum- expression converges. This is due the weight, that the factorials in the denominators accumulate. Bound for base-parameter s: However, in the case of infinite height we already know, that log(s) is limited to -e < log(s) < 1/e Differentability: In the general formula above we find the parameter for s as n'th power of its logarithm (divided by the n'th factorial). This simply provides infinite differentiablity with respect to s Differentiation with respect to the height-parameter m could be described when the eigensystem-decomposition is used (see below) (but may be I misunderstood the ongoing discussion about this topic completely - I kept myself off this discussion due to lack of experience of matrix-differentiation :-( ) In all, I think, that this description is undoubtedly the unique definition for the tetration of integer heights in terms of powerseries in s and x. ------------------------------------------------ For the continuous case my idea was, that it may be determined either by eigenvalue-decomposition of the involved matrices, or, where this is impossible due to inadmissible parameters s, by the matrix-logarithm. The eigenvalue-decomposition has the diasadvantage, that its properties are not yet analytically known, while matrix-logarithm is sometimes an option, if the eigenvalue-decomposition fails since it seems, that the resulting coefficients could be easier be determined. An example is the U-iteration U_s^(m)(x) = s^U_s^(m-1)(x) - 1, where s=e, which we discuss here as "exp(s)-1" - version. For this case Daniel already provided an array of coefficients for a series which allows to determine the U-tetration U_s^(m)(x) for fractional, real and even complex parameters m. These coefficients seem to be analytically derived/derivable, only the generating scheme was not documentd. However I provided a method how to produce these coefficients for finitely, but arbitrary many terms of the series for U_s^(m)(x)) by symbolic exponentiation of the parametrized matrix-logarithm, so at least that is a *basis* for the analysis of convergence and summability of the occuring series. Except for such borderline-cases I would prefer the definition of the continuous tetration based on the eigenanalysis of the involved matrix Bs. From the convergent cases -e

 Messages In This Thread Status of proofs - by Daniel - 08/24/2007, 06:56 PM RE: Status of proofs - by bo198214 - 08/24/2007, 07:16 PM RE: Status of proofs - by bo198214 - 08/24/2007, 07:23 PM RE: Status of proofs - by Daniel - 08/24/2007, 11:09 PM RE: Status of proofs - by bo198214 - 08/25/2007, 12:07 AM RE: Status of proofs - by bo198214 - 08/27/2007, 08:54 AM RE: Status of proofs - by Gottfried - 08/24/2007, 10:06 PM RE: Status of proofs - by Daniel - 08/24/2007, 11:02 PM RE: Status of proofs - by Gottfried - 08/29/2007, 06:07 AM RE: Status of proofs - by andydude - 08/29/2007, 05:04 AM RE: Status of proofs - by tommy1729 - 07/20/2010, 10:02 PM RE: Status of proofs - by bo198214 - 07/21/2010, 03:08 AM RE: Status of proofs - by tommy1729 - 07/21/2010, 10:39 PM

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