Taylor series of upx function

[Kouznetsov]

No. Until now, only the integral equation.

It it possible to compute Tetration in Mathematica using your method?

I would not recommend Mathematica for the numerical evaluation of the Cauchi integrals; it is slow. For the base e, the Taylor expansions are already calculated;
with these expanssions, the evaluation of tetration and its derivatives is fast.
You may use the code from
http://en.citizendium.org/wiki/Tetration...l.jpg/code

Is it possible to find an expression of, say \( (\text{sexp}_e x)' \) in 0?

Why I ask it? Because there is a property:
\( \ln f'(x)=f(-1)+f(0)+...+f(x-1) + \ln f'(-1) \)

for tetration with natural base \( \text{sexp}_e x \) and integer x (note also that \( \ln f'(-1)=\ln f'(0) \)).

I cannot yet answer. I hope to answer this question later.

did you discover some new functional/differential equations that tetration may satisfy, such as change-of-base formula, properties of derivatives/integral or relationship with other functions.

Not yet. Use the code above, try to guess some properties and try to prove them.

By the way, can you plot a graph of tetration \( \text{sexp}_{\sqrt 2} x \)?

Yes. Red dotted curve at
http://en.citizendium.org/wiki/Image:TetrationReal.jpg

Is it symmetric against y=-x line?

Wow! I did not check.. Perhaps, I should.