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 closed form for regular superfunction expressed as a periodic function Gottfried Ultimate Fellow Posts: 763 Threads: 118 Joined: Aug 2007 08/31/2010, 07:08 AM (This post was last modified: 08/31/2010, 11:28 AM by Gottfried.) (08/31/2010, 02:35 AM)sheldonison Wrote: And that would give you the four non-trivial terms I have closed form solutions for, although I haven't verified $a_5$, and wouldn't be surprised if it has a typo. - Sheldon Hi Sheldon - I recognize your coefficients. I came to the same coefficients, when I developed the eigensystem for the decremented exponentiation (dxp) which I called U-tetration. They occur as coefficients in the powerseries-expansion for the dxp (when including the iteration-height parameter h - which makes it also the superfunction for dxp). In http://go.helms-net.de/math/tetdocs/APT.htm I've described the procedure to solve for that coefficients based on the eigensystem/schröder-function-decomposition and gave some example coefficients-matrices. Here is another notation for the decomposition of your a_k coefficients in matrix-notation: [update] I updated the powers of L to simplify the matrix-columns - two errors corrected [/update] Code:```a0 = 1/0! /1              * 1   * [ 1                                                ] a1 = 1/1! /1              * L   * [ 1                                                ] a2=  1/2! /(L-1)          * L^2 * [ 2 + 1*L                                          ] a3=  1/3! /(L-1)(L^2-1)   * L^3 * [ 6 + 6*L + 5*L^2 + 1*L^3                          ] a4=  1/4! /(L-1)...       * L^4 * [24 +36*L +46*L^2 +40*L^3 +  24*L^4 + 9*L^5 + 1*L^6] ...```where I treat the coefficients in the brackets as matrix A: Code:```A = 1   .  .  .   .  .  .  ... 1   .  .  .   .  .  .  ... 2   1  .  .   .  .  .  ... 6   6  5  1   .  .  .  ... 24 36 46 40  24  9  1  ... ... ... ...``` Now the rows are known to me and match exactly the last columns of the A-coefficients-matrices in section "Coefficients of the powerseries for fractional iterates" There last column in A3 is [ 2 1 ], of A4 is [6 6 5 1], of A5 is [24 36 46 40 24 9 1 ] of A6 is [120 240 390 480 514 416 301 160 54 14 1] from where I am confident, that a5 in your case is explicitely Code:```a5 = 1/5! / (L-1)/(L^2-1)/(L^3-1)/(L^4-1)         * L^5 * [120 + 240*L + 390*L^2 + 480*L^3 + 514*L^4                    + 416*L^5 +301*L^6 +160*L^7 + 54*L^8 +  14*L^9 +1*L^10 ]``` However, I do not know the further exact relation of this to your approach, for instance I'm using general bases and also the log of the fixpoint (I called it "u", u=log(L) in this case) and its (fractional) h'th powers depending on the height h. I think I'll have to go through it step by step to find where and how your and my concepts match and where/how they differ in detail to make it possibly helpful for your considerations. Gottfried [update] Mike's and this msg seem to have crossed. Possibly the reference to the Faa di Bruno-formula is the more relevant/conclusive one for your problem [/update] Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread closed form for regular superfunction expressed as a periodic function - by sheldonison - 08/27/2010, 02:09 PM RE: regular superfunction expressed as a periodic function - by sheldonison - 08/28/2010, 08:44 PM RE: regular superfunction expressed as a periodic function - by tommy1729 - 08/28/2010, 11:21 PM RE: regular superfunction expressed as a periodic function - by sheldonison - 08/30/2010, 03:09 AM RE: regular superfunction expressed as a periodic function - by Gottfried - 08/30/2010, 09:22 AM RE: regular superfunction expressed as a periodic function - by tommy1729 - 08/30/2010, 09:41 AM RE: regular superfunction expressed as a periodic function - by tommy1729 - 08/30/2010, 09:46 AM RE: regular superfunction expressed as a periodic function - by Gottfried - 08/31/2010, 08:34 PM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/03/2010, 08:19 PM RE: closed form for regular superfunction expressed as a periodic function - by sheldonison - 09/05/2010, 05:36 AM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/05/2010, 04:45 PM RE: closed form for regular superfunction expressed as a periodic function - by sheldonison - 09/07/2010, 03:54 PM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/07/2010, 07:46 PM RE: closed form for regular superfunction expressed as a periodic function - by bo198214 - 09/08/2010, 06:03 AM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/08/2010, 06:55 PM RE: closed form for regular superfunction expressed as a periodic function - by bo198214 - 09/09/2010, 10:12 AM RE: closed form for regular superfunction expressed as a periodic function - by tommy1729 - 09/09/2010, 10:18 PM

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