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3 deleted 5 characters in body

Note: I asked this question in math.stackexchange but did not receive an answer

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 ${a_k}{k=0..\infty}$ (a_k ) _{k=0..\infty} $$using the double sum$$ \begin{array} {lll} s &=& \sum{r=0}^{\infty} 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} $$After arriving at the term -x e^{-x}  I've a vague impression I should have seen this transformation; but even if: I cannot remember. On the other hand - this summation-procedure is powerful, so this transformation is possibly interesting in more general use. Question: Does someone know this transformation and/or can provide a source where I can read more about it? 2 Formatting Note: I asked this question in math.stackexchange but did not receive an answer Background: In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-summation method. Beginning indexes at zero (r for "row", c for "column") the entries of M are$$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?

1

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

Note: I asked this question in math.stackexchange but did not receive an answer

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
$$s = \sum {r=0}^{\infty} a_r = \sum_{r=0}^{\infty} ( a_r \sum_{c=0}^r m_{r,c} ) = \sum_{c=0}^{\infty} ( \sum_{r=0}^{\infty} a_r m_{r,c} )= \sum_{c=0}^{\infty} b_c$$

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?