# Is that series-transformation known in the context of divergent summation?

Background: In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-summation method. Beginning indexes at zero (r for "row", c for "column") the entries of M are

$$m_{r,c}=\frac{eul(r,c)}{r!}$$ and $eul(r,c)=\displaystyle\sum_{k=0}^{c}(-1)^k \binom{r+1}{k} (c+1-k)^r$. (see for instance wikipedia)

The top-left segment of this (infinite, triangular) matrix is
$$\small{ \begin{array} {rrrrr} 1 & . & . & . & . & . \\\ 1 & 0 & . & . & . & . \\\ 1/2! & 1/2! & 0 & . & . & . \\\ 1/3! & 4/3! & 1/3! & 0 & . & . \\\ 1/4! & 11/4! & 11/4! & 1/4! & 0 & . \\\ 1/5! & 26/5! & 66/5! & 26/5! & 1/5! & 0 \end{array} }$$

The idea is, to sum a sequence $$\{a_k \} _{k=0..\infty}$$ using the double sum

$$\begin{array} {lll} s &=& \sum_{r=0}^{\infty} a_r &=& \sum_{r=0}^{\infty} ( a_r \sum_{c=0}^r m_{r,c} ) \\\ &=& \sum_{c=0}^{\infty} ( \sum_{r=0}^{\infty} a_r m_{r,c} ) &=& \sum_{c=0}^{\infty} b_c \end{array}$$

I was studying that summation with various sequences ${a_k}$ but I wanted to optimize the computation. For instance, if $\{a_k\}_{k=0..\infty} = q^k$ and thus define a geometric series with quotient q the $b_c$ are finite compositions of exponentialseries of $q$ and of powers of $q$: $$b_c = e^{q(1+c)}+\sum_{k=1}^c (-1)^k \frac{z^k+kz^{k-1}}{k!}e^z$$ where for sum-term $z=(c+1-k)q$ is simply a shortcut.

Writing the formula for the partial sums $$ps(q,c)= \sum_{k=0}^c b_k$$ this boils down to the following series-transformation:

$$ps(x,c) = e^{x(1+c)} \sum_{k=0}^c \frac{(c+1-k)^k}{k!}(-x e^{-x})^k$$ and we have in the limit $$\lim_{c\to \infty} ps(x,c) = \frac{1}{1-x}$$

After arriving at the term $-x e^{-x}$ I've a vague impression I should have seen this transformation; but even if: I cannot remember. On the other hand - this summation-procedure is powerful, so this transformation is possibly interesting in more general use.

Question: Does someone know this transformation and/or can provide a source where I can read more about it?

Here is an image showing the range of summability for the geometric series $- \infty \lt x \lt 1$ by the convergence of partial sums up to some index k:

• Not an answer, but in the context of Eulerian numbers and divergent series, you might be interested in math.ucsb.edu/~stopple/symmetriczeta.pdf – Stopple Mar 10 '11 at 16:56
• @Stopple: Hm, nice! I'll read it later in more depth. I'd done some heuristics concerning the Eulerian numbers but mostly lacking proofs. Perhaps you like go.helms-net.de/math/binomial_new/01_12_Eulermatrix.pdf as well. – Gottfried Helms Mar 10 '11 at 17:29
• It looks like it might be related to Borel summation. – Fabian Mar 10 '11 at 19:17
• You could try looking in Hardy's final book, Divergent Series - although this is rather old fashioned. – Zen Harper Mar 11 '11 at 7:13
• @Zen: thanks, seems I've to take a day in the lib. I hoped someone around could have known an immediate link/reference. I've read Hardy's book some years ago, but I think I've seen that "inverse Lambert-W"-term elsewhere in some article later; I think it is not in the book of K.Knopp, but possibly in one of its follow-up articles. sigh – Gottfried Helms Mar 14 '11 at 10:24