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 closed form for regular superfunction expressed as a periodic function Gottfried Ultimate Fellow Posts: 767 Threads: 119 Joined: Aug 2007 08/31/2010, 08:34 PM (This post was last modified: 08/31/2010, 08:40 PM by Gottfried.) Please excuse - this post contains some s But it's worth to notice how the bell-polynomials and my -for most fellows here: still cryptic - matrix-method are related. That I didn't see this earlier ----------------- In the mathworld-wolfram-link I find a short description of the Bell-polynomials first kind, fortunately with some example. $\sum_{k=0}^{\infty} \frac{B_k(x)}{k!}t^k = e^{(e^t-1)x}$ This expressed with my matrix-formulae is V(t)~ * dF(-1)*S2 * V(x) = exp( (exp(t)-1)*x) or using my standard-matrix for x->exp(x)-1 plus one intermediate step V(t) ~ * fS2F = V(exp(t)-1)~ write "y" for "exp(t)-1" V(y)~ * dF(-1) * V(x) = F(-1)~* dV(y)*V(x) = F(-1)~ * V( y*x) = exp( y*x) = exp( (exp(t)-1)*x) Or in one expression: V(t) ~ * (fS2F * dF(-1)) * V(x) = exp( (exp(t)-1)*x) (which I can recognize immediately to be correct because I'm extremely used to that notation) or the even simpler definition of the vector of Bell-polynomials: B(x) = S2*V(x) So now I see at least, how the Bell-polynomials of the *first kind* are related to my matrix-lingo. (So I should rename fS2F into "Bell" perhaps...) ---------------------------------- But then follows the Bell-polynomials of *second kind*. And there I'm lost again... No example, no redundancy... as if the reader could not do some error in parsing a complex formula... Is there possibly meant the iteration of the Bell-polynomials (=matrix-power of S2)? That would be simple then... Or, wait - looking at the *subscripted* "x" I think, that they are now coefficients for some arbitrary function developed as a powerseries with reciprocal factorials, ... and then... Well, this seems to be just the definition of what I found out myself at the very beginning of my fiddling with this subject and called it a "matrixoperator" for some function. With the additional property, that it is (lower) triangular (because of the missing x0 in the formula in mathworld), so for instance all my U_t-matrices for the decremented exponentiation (or "U-tetration to base t" in my early speak) were such collections of Bell-polynomials of second kind (but also the schröder-matrices, just all my lower-triangular matrix-operators which I worked with the last years and have the reciprocal factorial scaling ) So now - if we talk about the Bell-polynomials (first or second kind) and the symbolic representation in terms of the log(L) (or log(t)) then this is just what I solved in my earlier posted link (good to know)! Here it is again: http://go.helms-net.de/math/tetdocs/APT.htm . And Faa di Bruno/Bell-polynomials and "matrixoperators" is the same and one needs only read about one side of these notations... Amen - Gottfried 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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