• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Series for x-tetra-x andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 04/27/2009, 10:14 PM (This post was last modified: 04/28/2009, 12:12 AM by andydude.) I just found a nice "power series" for ${}^{x}x$. It's based on regular iteration, and I took the regular iteration power series for $\exp_x^y(z)$ and evaluated it at (z=1), then substituted (y=x) to obtain a power series for ${}^{x}x$. At this point, I noticed there were lots of logarithms, or in other words, it seemed to be of the form $\sum_{j\ge 0} \sum_{k\ge 0} A_{jk} x^j \ln(x)^k$ which is nice, but not very interesting. It was when I noticed that the power series expansion of $x^x$ about 0 also produces this kind of series that I thought the two could be combined to make a new kind of series. So I tried doing some linear algebra change-of-basis stuff but I forgot how, so I used "undetermined coefficients" instead. I want to discuss this more, but it will have to wait until this weekend. Here is the power series. $ \begin{tabular}{rl} (x + 1) \uparrow\uparrow (x + 1) & = 1 \\ & + x \\ & + x^2 (1 - x^x) \\ & + \frac{x^3}{2}(3 - 3 x^x) \\ & + \frac{x^4}{3!}(14 - 17 x^x) \\ & + \frac{x^5}{4!}(96 - 119 x^x - 12 x^{2x}) \\ & + \frac{x^6}{5!2}(1698 - 2013 x^x - 420 x^{2x}) \\ & + \frac{x^7}{6!4}(37448 - 41199 x^x - 15480 x^{2x}) \\ & + \cdots \end{tabular}$ [update]Fixed some coefficient signs[/update] Andrew Robbins « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post Perhaps a new series for log^0.5(x) Gottfried 3 4,707 03/21/2020, 08:28 AM Last Post: Daniel Taylor series of i[x] Xorter 12 23,700 02/20/2018, 09:55 PM Last Post: Xorter An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 24,955 01/14/2017, 09:09 PM Last Post: Vladimir Reshetnikov Complaining about MSE ; attitude against tetration and iteration series ! tommy1729 0 3,410 12/26/2016, 03:01 AM Last Post: tommy1729 2 fixpoints , 1 period --> method of iteration series tommy1729 0 3,471 12/21/2016, 01:27 PM Last Post: tommy1729 Taylor series of cheta Xorter 13 25,930 08/28/2016, 08:52 PM Last Post: sheldonison Derivative of E tetra x Forehead 7 15,530 12/25/2015, 03:59 AM Last Post: andydude [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried 33 67,146 03/27/2015, 11:28 PM Last Post: tommy1729 [integral] How to integrate a fourier series ? tommy1729 1 4,908 05/04/2014, 03:19 PM Last Post: tommy1729 Iteration series: Series of powertowers - "T- geometric series" Gottfried 10 25,501 02/04/2012, 05:02 AM Last Post: Kouznetsov

Users browsing this thread: 1 Guest(s)