Re: Existence of function

Am 16.11.2007 17:12 schrieb I.N. Galidakis:
hagman wrote:
[snip]

We only have an implicit functional equation for f. Worse, the
functional equation only works for the Naturals, hence we only know
the set {f(n):n \in N U {0}}.
I see.
So the problem at hand is apparently somewhat similar to
the transition from factorial to Gamma function.

That's right. Gamma was demonstrated and additionally it was proved that it's
the only log-convex function which extends the factorial.

So, in a sense, Gamma has passed from "non-existence" to "existence" in our
mental space, as soon as it was demonstrated.

f in tetration cannot be demonstrated, hence in my view it cannot obtain the
status of "existent", based only on numerical evidence.

I guess the question boils down to which mathematical objects "attain" the
status of "existent" and can be subsequently considered part of a
Mathematician's ammunition.

In my opinion, if an object cannot be demonstrated (using all available
"analytical" means), it "fails" to attain the status of "existent", whether it
really exists or not.
Ioannis,

I've a small handout for the U-tetration x->exp(x)-1 (this is not
1-1 translatable to other bases b^x-1 currently)
see http://go.helms-net.de/math/tetdocs/CoefficientsForUTetration.htm

It shows the problem of divergence of the sequence of coefficients,
but also shows, that -at least for a certain range- these divergences
can be overcome by techniques of divergent summation and thus allow
to formulate series and assign values to them.

For other bases the entries in U are also (logarithmic) series, and
I don't know the evaluations for them (but possibly this is also
manageable with a deeper analysis)

Gottfried

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Gottfried Helms, Kassel
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