Please excuse - this post contains some <aaarrrgggh>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 <arrggh!>)
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 -
<cough>
Gottfried
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 <arrggh!>)
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 -
<cough>
Gottfried
Gottfried Helms, Kassel