In proceeding with an older discussion of a summation-procedure for divergent series using the matrix of Eulerian numbers I came to a rather general formulation but cannot/do-not-know-how-to make sure, that the limit of some partial sums is indeed a correct expression.
What I got so far is:
let
$$f(x) = a_0 + a_1 x + a_2 x^2 + ... \tag1 $$
be the formal power series of some suitable function and
$$g_0(x) = a_0 + a_1 x/1! + a_2 x^2/2! + ... \tag2 $$ its "$E_0$"-transform. We'll discuss only functions $f(x)$ whose $E_m$-transforms $g_m(x)$ have a nonzero range of convergence, so for instance geometric series.
Using the decomposition of the matrix of Eulerian numbers I arrive at the following general scheme of further transforms
$$g_1(x) = a_0 \cdot b_{0,1} + a_1 \cdot b_{1,1} x/1! + a_2 \cdot b_{2,1} x^2/2! + ... \\
g_2(x) = a_0 \cdot b_{0,2} + a_1 \cdot b_{1,2} x/1! + a_2 \cdot b_{2,2} x^2/2! + ... \\ ... \tag3$$
where the $b_{r,c}$ are the coefficients of the Eulerian matrix. It is crucial here, that the $g_n(x)$ all are only finite compositions of derivatives of $g_0(x)$ at $x,2x,3x,...$ and so on.
The hypothesis is that in the limit $f(x) = g_0(x)+g_1(x)+g_2(x)+... $.
The key-idea of that summation-procedure was, that a powerseries $f(x)$ which might be divergent for some $x$, might have a convergent powerseries $g_0(x)$ and likewise $g_1(x),g_2(x),...$ This is for instance true if $f(x)$ is a geometric series and for the infinite interval $ - \infty < x <1 $ the procedure gives the correct results. I can even sum $ f(x) = 0! - 1!x + 2!x^2 - 3!x^3 + ... - ...$ to the correct value which suggests, that this summation procedure is possibly as powerful as the Borel-summation.
But now I want to discuss it for more generality. Evaluating formally the occuring partial sums $$p_n= \sum_{k=0}^n g_k(x) $$ I arrive at the following scheme:
$$ \begin{array}{} p_0(x) &=& g_0(1x) \\ p_1(x)& =& g_0(2x) &-& (1x)g_0^{(1)}(1x) \\ p_2(x)& =& g_0(3x) &-& (2x)g_0^{(1)}(2x) &+&(1x)^2 \cdot g_0^{(2)}(1x)/2! \\ ... \\ \end{array} \\ \begin{array}{} p_n(x)& =& \sum_{k=0}^n (-(n+1-k)x)^k {g_0^{(k)}((n+1-k)x) \over k!} \end{array} \tag4 $$ and the hypothese $$ f(x) = \lim_{n \to \infty} p_n(x) \tag5$$ The strange thing is, that having that description by derivatives and reciprocal factorials, this looks like some turned-around ( or: "mirrored") taylor-series but possibly by this simple property it is known, that this composition of derivatives is also valid for at least some functions $f(x)$ (or: just invalid/nonsense).
Q1: Is the final hypothese $$ f(x) = \lim_{n \to \infty} p_n(x)$$ $ \qquad \qquad $true, at least for some class of functions?
Q2: and how can I show that?