Timeline for A curious series related to the asymptotic behavior of the tetration
Current License: CC BY-SA 3.0
23 events
| when toggle format | what | by | license | comment | |
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| Apr 13 at 19:32 | comment | added | Vladimir Reshetnikov | @TymaGaidash Yes, it is wanted very much. Thanks! | |
| Apr 7 at 16:01 | comment | added | Тyma Gaidash | @VladimirReshetnikov There may be a closed form for $a_n$ in $c_x=\sum\limits_{n=1}\frac{a_n}{n!} x^n$ in the form $a_n=\sum\limits_{m_1}\dots\sum\limits_{m_n}f(m_1,\dots,m_n)$ with finite sums. Is it wanted? | |
| Jun 16, 2017 at 8:55 | answer | added | Gottfried Helms | timeline score: 2 | |
| S Jun 15, 2017 at 18:55 | history | bounty ended | Vladimir Reshetnikov | ||
| S Jun 15, 2017 at 18:55 | history | notice removed | Vladimir Reshetnikov | ||
| Jun 14, 2017 at 8:09 | answer | added | Gottfried Helms | timeline score: 9 | |
| Jun 14, 2017 at 6:19 | answer | added | Alexey Ustinov | timeline score: 5 | |
| Jun 10, 2017 at 2:48 | comment | added | Vladimir Reshetnikov | Apparently, the coefficients in the series $(6)$ can be generated by this quite simple Mathematica function. Can we extract a readable formula from it? | |
| Jun 9, 2017 at 17:46 | comment | added | Gottfried Helms | Vladimir - ah, now I see. Well, unfortunately I'm short now in time today, and do not know, whether I'll find time and energy in the evening (I have to give a course at weekend). Perhaps I feel free at saturday or sunday evening for this. | |
| Jun 9, 2017 at 16:45 | comment | added | Gottfried Helms | (???) - My reproduced series expansion for definition (6) is taken from a symbolic analysis (indepently from the value in $\lambda$ / keeping $\lambda$ indeterminate) . So I'm possibly missing something in your question? | |
| Jun 9, 2017 at 12:15 | comment | added | Gottfried Helms | I could now reproduce your series (6) using the Schröder-mechanism in symbolic algebra, and fully in rational coefficients at the powers of $\lambda$ (in Pari/GP) . The coefficients remain rational even when diagonalizing the Carleman-matrix for $a^x$ (after conjugacy-map) , because after the conjugacy-map the Carlemanmatrix is triangular with powers of $\lambda$ (which of course evaluate to irrational values when the symbolic notation is resolved) and allows an eigensystem in rational numbers. So your coefficients in (6) are exact/correct. But I have not yet a simpler pattern for them... | |
| Jun 9, 2017 at 4:05 | comment | added | Vladimir Reshetnikov | The series reversion of $(6)$ gives a series whose coefficients apparently have alternating signs and monotone absolute values: $\frac{c}{1!}-\frac{c^2}{2!}+\frac{5c^3}{3!}-\frac{24c^4}{4!}+...$ It seems more likely to find a general formula for them. The absolute values of numerators can be found here. | |
| Jun 9, 2017 at 3:00 | comment | added | Vladimir Reshetnikov | Related OEIS entries: oeis.org/A198094, oeis.org/A260691, oeis.org/A277435 | |
| S Jun 9, 2017 at 2:57 | history | bounty started | Vladimir Reshetnikov | ||
| S Jun 9, 2017 at 2:57 | history | notice added | Vladimir Reshetnikov | Draw attention | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Jan 24, 2017 at 6:52 | comment | added | Gottfried Helms | Would you mind to explain how you arrived at the series for $c_\lambda$? Tinkering with my representations via the Schröderfunction I could not yet reproduce the coefficients and arrived at different coefficients instead. | |
| Jan 24, 2017 at 1:24 | comment | added | Gottfried Helms | If the Schröder-formula in the previous comment is really meaningful, then the maximum of the absolute value of $c_\lambda$ seems to occur near $\small \lambda \approx \log(1.98129024000)$ giving $\small c_\lambda \approx -0.632423221806$ | |
| Jan 24, 2017 at 0:51 | comment | added | Gottfried Helms | Just for the record, perhaps the taylor-series comes from here: I get it as the value of the schröder-function by: $c_\lambda=\sigma(1/t-1)\cdot t$ where I denote with $t$ the fixpoint, in the example $t=2$ Here the term $(1/t-1)$ results from the conjugacy in the Schröder-mechanism, where we write $ \exp_a^{\circ h} (z) $ for the h'th tetrate beginning at $z$ and the Schröder-mechanismn works on $ \exp_a^{\circ h} (z) = \sigma^{-1}(\lambda^h \cdot \sigma(z/t-1))+1)\cdot t $ and $\;^n a= \exp_a^{\circ n}(1) $ *(Just to compare, for $t=1.5,a=t^{1/t}$ I get $c_\lambda \approx -0.448243486$ )* | |
| Jan 23, 2017 at 23:58 | history | edited | Vladimir Reshetnikov | CC BY-SA 3.0 |
added 54 characters in body
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| Jan 23, 2017 at 23:48 | comment | added | Vladimir Reshetnikov | Here is $c_\lambda\approx0.632098661...$ for $a=\sqrt2 \, (\lambda=\ln 2)$, computed with $30000$ decimal digits of precision: goo.gl/gkNjkD | |
| Jan 23, 2017 at 23:37 | history | asked | Vladimir Reshetnikov | CC BY-SA 3.0 |