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While reading through the Citizendium article on tetration, iterated exponentation, I came across a power series approximate of tetration. The article said that it got the series coefficients from cauchy’s integral formula for the nth derivative of a function. There are a few questions I have. Firstly how did they get the power series coefficients given only the recursive definition for tetration? Secondly I’m familiar with the fact that you can determine the nth iterate of the exponential function through Carleman matrices and then plug 1 into the function and obtain tet(n) for all n, will this function converge to the same function represented through the power series expansion of tetration?

daw
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Anthony Corsi
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    I would not call tetration (a term that I utterly detest) the "fourth (hyper-)operation above exponentiation". Rather, I would call it the first operation above exponentiation, specifically "iterated exponentiation", and in particular, "continuously iterated exponentiation". You might want to look at this 1960 paper of Erdos & Jabotinsky, and later ones that reference it: https://static.renyi.hu/~p_erdos/1960-07.pdf. – Dan Asimov Oct 13 '23 at 02:33
  • Your questions are probably more suitable to the discussion page for the article you link to. – Ryan Budney Oct 13 '23 at 08:00
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    Additional to Daniel Asimov's comment, I would always say: "... tetration, the *expected* fourth..." or "... tetration, the *attempted* fourth..." because it is still not established, that such a function is meaningfully constructible for the general case at all. If I recall correctly, even in the citizendium article the author has let it as a hypothese (although his attempt to provide evidence for it). Btw., I would not comment this way in MSE, but in a forum like MO I'd like to see what I think is more professional style. – Gottfried Helms Oct 13 '23 at 11:58
  • According to the Wikipedia page https://en.wikipedia.org/wiki/Tetration there is a link to a paper which supposedly proves that tetration is holomorphic: https://link.springer.com/article/10.1007/s10444-017-9524-1 – Max Horn Oct 13 '23 at 19:27
  • In my papers focused on (strong) integer tetration, I use both "tetration" and "hyper-4" as synonyms. Anyway, my favourite extension of hyper-4 to the complex plane is the one provided by William Paulsen (see https://link.springer.com/article/10.1007/s10444-018-9615-7). The discussion is still open on tetrationforum, with some interesting outcomes as the following thread https://tetrationforum.org/showthread.php?tid=1762. – Marco Ripà Oct 14 '23 at 00:27
  • I thought Kneser's Tetration gave a unique solution of extending tetration as a holomorphic function the complex plane (minus the real axis) and so its power series at various points became the canonical power series for tetration. Clicking around on @Marco's tetration forum link suggests that there is still activity to come up with different fractional analogs of tetration. Is there something undesirable about Kneser's solution that motivates people to keep looking for other forms of tetration with different power series? See here: https://en.wikipedia.org/wiki/Tetration#Complex_heights – Sidharth Ghoshal Oct 14 '23 at 02:26
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    @SidharthGhoshal - there exists a provisorical software solution for the Kneser-Ansatz; but the author (S.Levenstein) conceded, that its correctness needs external verification. It's true that in the tetrationforum-biotop this is much accepted - even in its extension to other bases than $e$ and even to complex values of its arguments; and I don't know in which way Kouznetsov(citizendium) , Kouznetsov & Trappman's, Paulsen's, and recently J.D.Nixon's "beta" method are related to this. The Levenstein's implementation is impressive and may have the best chance to come out as canonical solution. – Gottfried Helms Oct 14 '23 at 17:00
  • Gottfried Helms: The case of continuous iteration of an analytic function f : U —> ℂ near a fixed point (say at f(0) = 0) is nicely covered by Königs's Theorem: If |f'(0)| is neither 0 nor 1, then there exists a holomorphic family {f_t | t ∈ V} (where V is an open neighborhood of the origin) satisfying f_0 = f and f_s o f_t = f_(s+t) for all s and t with |s| < epsilon, |t| < epsilon) where s and t are complex. (For some epsilon > 0.) In many cases, the iteration and its properties are satisfied for all s, t complex. – Dan Asimov Oct 14 '23 at 23:06
  • (continued) In most of those cases, there exist more than one way to define this. Thus ultimately depends on a choice of one specific branch of the complex logarithm. – Dan Asimov Oct 14 '23 at 23:11
  • @DanAsimov - thanks for your comment! Is by the analysis of Koenigs something about the order of epsilon known? By some (little educated) considerations on possible trajectories over periodic points I came to the guess that that epsilon might be very small. (I've done one or two msg's on MO concerning periodic points but couldn't meaningfully resolve my doubts/establish a sound understanding) – Gottfried Helms Oct 15 '23 at 17:32
  • @GottfriedHelms the "beta" method is not analytic. Tommy's improvement (of the beta) by the gaussian method is analytic. But is probably not equal to the Kneser. Tommy's method with $\exp(x) - \exp(-3/5 x)$ might be the same as Kneser. – mick Nov 20 '23 at 22:24
  • @mick - as those concepts are not easily to decode practically: would you mind to give some numerical evidence for the two topoi you mentioned? Numerical comparision of the Kneser (by Levenstein?) and the Nixon's Beta and/or Tommy's solutions? (At least for the sake of having redundance when I should try myself to implement&check) That yould be a great service... – Gottfried Helms Nov 29 '23 at 12:12
  • not so easy Gottfried , but it was somewhat in the forum – mick Nov 29 '23 at 12:28
  • I've no knowledge about this Cauchy-Integral method used by D. Kouznetzsov in the citizendium-article so I can't say whether this is helpful: There is a working paper of Trappmann, Robbins, Chernykh of Apr 2009 titled "5+ methods for real analytical tetration" where they attempt to explain in short this Cauchy-integral method for the real case, chapter 9. Maybe this is on arxiv, if not then only in the tetrationforum at https://tetrationforum.org , see "resources & Wiki". Hope it is useful for your question... – Gottfried Helms Aug 17 '24 at 05:06
  • Moreover, there exist a next working-paper of Trappmann with the author of the citicenrium article (Kouznetsov) with the same title, but of Jun. 2010. The Cauchy-integral method is now in chap 7 and looks a bit better explained. Still I don't see myself how to generate the coefficients of the powerseries by this but perhaps it is helpful for someone more familiar with the Cauchy-integral. – Gottfried Helms Aug 17 '24 at 05:16
  • For those intrested ; tommy's method : https://sites.google.com/site/tommy1729/tetration – mick Oct 01 '24 at 22:47

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On your 2nd question:

Secondly I’m familiar with the fact that you can determine the nth iterate of the exponential function through Carleman matrices and then plug 1 into the function and obtain tet(n) for all n, will this function converge to the same function represented through the power series expansion of tetration?

Well you consider the Carleman-method only for integer heights (=integer powers of matrix). There is no solution known for computing fractional heights this way - we needed an idea, how to diagonalize and from this fractionally power the Carlemanmatrix (which is of infinite size, btw.).

The only viable attempt so far is to recenter the power-series of the exponentiation about its fixpoint and create the according Carlemanmatrix - but firstly that fixpoint is complex, and secondly there are infinitely many such fixpoints, and from this all those fixpoints provide different fractional powers/solutions for fractional heights. Moreover, the solution for the fractional powers give then complex values even for real heights of iteration - so we might say: the Carleman-method is (as far as we know today) not feasible for this (type of) function. (For functions with real fixpoint the diagonalizing of the according Carlemanmatrix can be done in real values and we can perhaps/sometimes derive results in real numbers, see "Schroeder-" or "Koenigsfunction").

So, the comparision with the results by the citizendium-powerseries shall give identity for the integer heights (all methods should agree on integer heights!) but not with the fractional heights .

p.s. The Carleman-ansatz deals with common power-series of the form $f(x)=K+ax+bx^2+cx^3+...$ Since not too long I'm having doubts, whether this ansatz is sufficient at all for -for instance- $f(x)=\exp(x)$ or --- whether we need an extension for the handling of Laurent-series (like we need this for the analytical acceess of the $\zeta()$-function). But aside of a couple of attempts to make this operational there is nearly nothing at all I know of this, except the one attempt by Eri Jabotinsky in the 70ies-90ies. However I don't see this attempt usably supplied with concept and practical operationalizing, so I couldn't take much of the (one) article I've found so far...

Gottfried Helms
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  • The limit of some truncated carleman matrices however seems to work. – mick Oct 01 '24 at 22:43
  • How does one create the Carlemann matrix series for the recenter of the exponential function? I know how it can be done with the regular exponential with an infinite series, but on wikipedia I have only seen it done with Taylor series. Would you have to use a Laurent series? – Anthony Corsi Oct 02 '24 at 01:23
  • @AnthonyCorsi - first a question: what do you mean with "with the regular exponential ... with an infinite series" in contrast to "with Taylor series": I thought we're talking about Taylor-series when discuss the regular exponential? (... cntd...) – Gottfried Helms Oct 02 '24 at 06:35
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    (...cntd...) Second is just some guess how to overcome the problem of the zero-column in the Carleman-matrix, when we, for instance want to invert $(B - I)$ where $B$ is the Carlemanmatrix for $\exp()$ (as attempted by Peter Walker and in the forum by A. Robbins). A small discussion may be instructive about what I mean: https://go.helms-net.de/math/tetdocs/ProblemWithBellmatrix.pdf . Here I speculate with introducing a $1/x$-term into the power series - but without arriving at a conclusion. – Gottfried Helms Oct 02 '24 at 06:37
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    @mick - the concept ot the limit of some truncated carleman matrix - well as P. Walker already mentioned: the series seem to converge to something reasonable - but applying this series gives values for the fractional iterates which seem to be different to another method he prefers in his article. So he does not conjecture that this method would be "the meaningful one". (... cntd ...) – Gottfried Helms Oct 02 '24 at 06:42
  • (... cntd ... @mick) By comparison with increasing sizes up to 96x96 it seemed that the results might converge to that of the Kneser-implementation in the Forum - perhaps with a tiny systematic deviation. But note that J.D.Fox in the tetration-forum extended the truncation size up to 700x700 and still has not been satisfied with the result if I remember correctly. – Gottfried Helms Oct 02 '24 at 06:45
  • @GottfriedHelms (in your paper) why are you adding all elements of bell or carleman/jabotinsky matrices for exp ? That is not the same as taking a power of it ? – mick Oct 02 '24 at 22:53
  • @mick - I like to look at things in various perspectives; the difficulty of a Carleman-matrix which adds-like-powers, for instance, made me look at many properties, even some that I had not seen in articles. The Carleman-matrix for exponentiation exhibits similarily some seemingly oddnesses, let's only think of the difficulty of diagonalization. Now I look at that matrix this and that way and try to explain any oddity that occurs to me - the idea of summing columnwise vs rowwise and see what happens is not a homework from some textbook, just to analyze possible reasons for that oddity... – Gottfried Helms Oct 03 '24 at 09:48