# Tetration Forum

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Hi -

one remark at the beginning. I don't have another name than "powertower" for a singe tetration-term, so I still use this word here.

As you rememeber, at my initial interest were not the deep hardcore properties of the tetration-function, but rather some series, evaluated in my more general compilation of relations of important number-theoratical coefficients-matrices like binomials, stirling-numbers, bernoulli-numbers and the like (where tetration popped up as an iterative application)
I guess, it is of worth, to add to the knowledge-base also that pieces about series of tetration-terms, aka powertowers.

Here are two conjectures about identies of series; they are analogons to the known geometric series (if the tetration-exponent/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 threads-list
Hey Gottfried,

that sounds really interesting, however give me some time to thoroughly read it.
bo198214 Wrote:Hey Gottfried,

that sounds really interesting, however give me some time to thoroughly read it.

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
@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 umbral-calculus style of proof. From what I understand umbral calculus is making transitions that are not rigorously true, but strictly syntactic substitutions. For example:


\begin{array}{rl}
e^x & = \sum_{k=0}^{\infty} \frac{1}{k!} x^k \\
f(e, x) & = \sum_{k=0}^{\infty} \frac{1}{k!} f(x, k)
\end{align}

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
andydude Wrote:@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 umbral-calculus style of proof. From what I understand umbral calculus is making transitions that are not rigorously true, but strictly syntactic substitutions. For example:


\begin{array}{rl}
e^x & = \sum_{k=0}^{\infty} \frac{1}{k!} x^k \\
f(e, x) & = \sum_{k=0}^{\infty} \frac{1}{k!} f(x, k)
\end{align}

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

Hi Andrew -

thanks for the comment!
I know a bit about umbral-calculus from the context of Bernoulli-numbers, 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 math-department 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 computer-science to social-assistance, so I'm doing number-theory 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 eta-function as replacements of the infinite sums of cofactors.

A notation

$\hspace{24} V(x)\sim * Bs = V(y)\sim$

is nothing more than a shortcut for the explicite, well known exponential-series for all consecutive powers of y; say in a column c of the result vector

(1) $\hspace{24} y^c = \sum_{r=0}^{\infty} x^r * log(s)^r * c^r/r!$

where from the elementary properties of the exponential-series

$\hspace{24} y^c = (s^x)^c$

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:
$\hspace{24} \begin{eqnarray}
y_0^c &&=&& \sum_{r=0}^{\infty} x_0^r * log(s)^r * c^r/r! \\
y_1^c &&=&& \sum_{r=0}^{\infty} x_1^r * log(s)^r * c^r/r! \end{eqnarray}$

and

$\hspace{24} y_0^c - y_1^c = \sum_{r=0}^{\infty} (x_0^r-x_1^r) * log(s)^r * c^r/r!$

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
$\hspace{24} (x_0^r - x_1^r + x_2^r - + ... ) = (0^r - 1^r + 2^r - 3^r...)$
and that
$\hspace{24} (0^r - 1^r + 2^r - 3^r...) = eta_0 ( r)$
is interchangable for the linear combination of x_k.

What we have is then the double-sum for a column c of the result-vector

$\hspace{24} y^c = \sum_{n=0}^{\infty} \sum_{r=0}^{\infty} ((-1)^n *n^r) * log(s)^r * c^r/r!$

which is, after interchanging the order of summation
(2) $\hspace{24} \begin{eqnarray}
y^c &&=&& \sum_{r=0}^{\infty} log(s)^r * c^r/r! * \sum_{n=0}^{\infty} (-1)^n*n^r \\
y^c &&=&& \sum_{r=0}^{\infty} log(s)^r * c^r/r! * eta_0(-r) \\
\end{eqnarray}
$

The lhs is now the sum
$\hspace{24} y^c = \sum_{n=0}^{\infty} (-1)^n*(s^n)^c$
but the interesting result is only in the column where c=1 so
$\hspace{24} y = \sum_{n=0}^{\infty} (-1)^n*(s^n) = s^0 - s^1 + s^2 ...$

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 eta-values of appropriate exponents.
For the base-parameter 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 Euler-summed.

This all is nothing else than to rewrite in serial-notation, what is implicite when using the notation of matrix-multiplication.
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 math-departement :-))

Regards -

Gottfried
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 G-function 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.
(08/24/2007, 02:44 PM)Gottfried Wrote: [ -> ]... Here are two conjectures about identies of series;.. Gottfried
http://go.helms-net.de/math/pdf/Tetratio...ort.pdf%20
causing diagnistics

2. Luann Cole urgently needs to find bo198214
If anybody see him, let Bo communicate Luann, please.
(02/03/2012, 03:06 PM)Kouznetsov Wrote: [ -> ]
(08/24/2007, 02:44 PM)Gottfried Wrote: [ -> ]... Here are two conjectures about identies of series;.. Gottfried
http://go.helms-net.de/math/pdf/Tetratio...ort.pdf%20
causing diagnistics

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 matrix-method 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 Ramanujan-summation, where we need to consider one additional integral-expression for the validity of the divergent Ramanujan-summation. But I did not find yet an appropriate expression for this in the case of the iteration-series 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
(02/03/2012, 04:47 PM)Gottfried Wrote: [ -> ]The link is http://go.helms-net.de/math/pdf/Tetration_GS_short.pdf
Thank 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 iteration-series of powertowers/tetration. ..
?
Does your method allow to evaluate tetration faster (or more precise) than my one?
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?
Hello Dmitri -

(02/03/2012, 06:19 PM)Kouznetsov Wrote: [ -> ]
(02/03/2012, 04:47 PM)Gottfried Wrote: [ -> ]..But I did not find yet an appropriate expression for this in the case of the iteration-series of powertowers/tetration. ..
?
My conjecture, based on standard matrix-identities which I assumed could be extended to the case of infinite size, was the following. I constructed the doubly-infinite series of powertowers of increasing/decreasing height, so for the index/height h of -infinity to +infinity. By the matrix-identities (involving the von-Neumann-series of the according Carlemanmatrix and its inverse) I expected that the sum of that doubly-infinite series was always zero.
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 Ramanujan-summation.
I have described this more precisely at
http://go.helms-net.de/math/tetdocs/Tetr...roblem.pdf

Some more introductory remarks to the concept of iteration-series are in the introductory remarks at my tetration-homepage at http://go.helms-net.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 "Carleman-matrix" which I did not know when I came across the problem of tetration and my general matrix-approach to some number-theoretic problems.
So it has the known deficients of the Carleman-matrix-approach:
a) there is nothing known yet which would make the Carlemanmatrix-approach 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 complex-valued 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!