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The series 0{b}h/0! - 1{b}h/1! + 2{b}h/2! - ... + ...

In my earlier posts I considered three types of tetrational series

Tetra-geom-series : Example for h=2:

Tetra-eta-series : Example for h=2:

Tetra-series : Example:

These series have all certain ranges of convergence or at least of conventionally manageable divergence like for instance 1-2+3-4... .

It is interesting, how these series could be reformulated in the matrix-operator concept. Since the matrix-approach gives some matrix expression for each term, the alternating sums can formally be expressed by linear combinations of the involved matrices or vectors, and using assiciativity allows then sometimes to replace the infinite sums of matrices or vectors by one matrix or vector alone, whose entries are analytically known.

It was then interesting, that the involvement of infinite series of matrices/vectors is only correct for "one direction" - either a "closed form"(like it is known for the scalar geometric series) of infinite series for k=0 to inf could be verified by crosscheck with serial summation of the scalar expressions but not the series of k=0 to -inf or the converse. We need a correction-term for the incorrect closed-form-expressions, which seems to include a sinusoidal function.
A similar effect is known for scalar lambert-series, which usually are restricted to a "finite to infinity"-range of index.
I've not yet fully understood the effect - but maybe there will be some progress here.

Note, that in all mentioned series no fractional iterate is requested, so we deal only with integer iterates. This seems to prevent some problems with the fixpoint-shifts. Also the diagonalization of the matrix-operators to get their powers is not really needed.

Here I want to introduce some first results for the
Tetra-exp-series : Example for h=2:

The series of height h=1 is simply the exponential-series
AE(b,1) = b^0/0! - b^1/1! + b^2/2! - ... + ...
which converges for all b.

In the matrix-notation it is, with the operator Bb for base b
Y~ = V(0)~*Bb - V(1)~/1!*Bb + V(2)~/2! *Bb - ... + ...
AE(b,1)= Y[1] // the second scalar entry in Y

This can then be expressed as

Y~ = (V(0)/0! - V(1)/1! + V(2)/2! - ... + ...)~ *Bb
Y~ = exp(-1)* X~ * Bb
AE(b,1) = Y[1]

where X~ is a rowvector.

The entries of X can be determined analytically and due to the dominance of the factorial in the denominator we get convergent expressions for all entries.

The sequence of entries in X are an interesting sequence; it is known as "Rao Uppuluri Carpenter-numbers" in the OEIS with id A000587
and are

X~ = [1, -1, 0, 1, 1, -2, -9, -9, 50, 267, 413, -2180, ... ]

Some example-computations with different bases show, that indeed the matrix-formula gives the correct results for height h=1.

Next we use h=2

Here we have to restrict ourselves to some range for the bases such that the sum converges or is conventionally summable by serial summation of the scalar representation of individual terms, for instance |b|<=1. The matrix-representation uses simply an integer power of Bb:

AE(b,2) = exp(-1)*X~ * Bb^2

and for the given range some examples confirm this method of summation.
For b=0.9 I got

AE(b,1) = b^0/0! – b^1/1! + b^2/2! +... - ... = 0.406569659741
AE(b,2) = b^b^0/0! – b^b^1/1! + b^b^2/2! +... - ...= 0.327420772888
AE(b,3) = b^b^b^0/0! – b^b^b^1/1! + .. - ... = 0.334950080400

with both methods.

However, the extension for b excessing the range of convergence (or conventional summability) using the matrix-method seems smooth and again seems to provide sort of "analytical continuation" (I'm not really used to the confirmation of such analytical continuation) but things look very promising.

It is also very intriguing, that we need only one constant vector X~ for all heights h; the height h is only expressed by powers of Bb. This indicates a strong relationship between all these series over their different heights.

So far just this short note.

Gottfried Helms, Kassel

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