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From: Gottfried Helms on 3 Feb 2010 05:50
Hi -

some monthes ago I fiddled with the subject of
divergent summation using matrix-transformations,
namely using the matrix of Eulerian-numbers

E =
1 0 0 0 ...
1 0 0 0 ...
1 1 0 0 ...
1 4 1 0 ...
1 11 11 1 ...
... ...

If we use the matrix to write the double-sum (r=row,c=column-index)

Su(x) = sum_{r=0..inf} ( sum_{c=0..r} x^r*E[r,c] )

we can evaluate this in two ways:
a) first sum along each row, then sum that rowsums

Su(x) = sum_{r=0..inf} x^r ( sum_{c=0..r} E[r,c] )
= sum_{r=0..inf} x^r c!

b) first sum along each column, then sum that column-sums

Su(x) = sum_{c=0..inf} ( sum_{r=0..inf} x^r E[r,c] )
= sum_{c=0..inf} f_c(x)

where f_c(x) is a columnspecific, but finite composition of
geometric series and derivatives with parameter x.

The notation in a) gives a powerseries with convergence-radius zero;
the notation b) gives a series, whose terms result from geometric
series, which, for x=-1 are all divergent. But because geometric
series and their derivatives have a simple analytic continuation
it appears, that we can base a summation on this matrix-transformation
and we get the correct value even for Su(-1). (giving the Gompertz-
constant of ~ 0.59... ).

I have three questions ( I'll post a second mail with the third question)

1) The method is the same as the naive method of computing
Euler-means/binomial-transformation, only we use the Eulerian-matrix
instead of the Pascal-Matrix/matrix of binomials.
But while the Euler-means (or Euler-summation) can only sum
geometric series and no series of stronger growth, using the Eulerian
numbers we can even sum (alternating) factorials.

Was that approach discussed anywhere? (I've not seen this -for
instance- in the K.Knopp-monography on series)

2) Once we see, that there is simply another (triangular) matrix
besides the Pascal-matrix (not even too exotic!), which allows
a simple series-transform for divergent summation, we can ask:
how must a matrix look like to be able to sum still more divergent
series, say series of squares of factorials, or series like
S(x) = x - x^2 + x^4 - x^8 + x^16....
and such.
But when I tried to construct appropriate matrices for such
series I stuck.
Is there a limit for the power of the matrix-method (if I restrict
myself to lower triangular matrices first)? And if there is such
a limit, how can it be described?

Gottfried Helms

------------------------------------

I have a short discussion of 1) in
http://go.helms-net.de/math/binomial_new/01_12_Eulermatrix.pdf

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