A tiny base-dependent formula for tetration (change-of-base?)

[Gottfried]

Sorry for being little out of touch, to me "upping " things one step looks very interesting in general ( because of log-Poisson distribution):


\( e^{\lambda^k /k!*e^{-\lambda} \)

Can we not replace \( \lambda^k/k! \) by any function \( f(\lambda) \) which has a powerseries expansion, as that expansion would usually contain something like \( c* \lambda^k/k! \) ? Of course, coefficients at powers of \( \lambda \) can be also anything else. I can only see so far that terms of powerseries expansion has to be put into "double exponentiation operator" one by one:

\( e^{f_k (\lambda)*e^{-\lambda}} \)

Then the summation of terms of powerseries has to happen in exponent, and than we end up with :

\( e^{f(\lambda)*e^{-\lambda}} \)

Perhaps than we can replace \( f(\lambda) = f (T(z)) \) and see what happens. In case \( f( \lambda) = \lambda =T(z) \) we get

\( e^{T(z)*e^{-T(z)}=e^z \)

\( T(z) \) is Euler tree function.

if now \( z=\ln w \) then

\( e^{T(\ln w) * e^{-T(\ln w)} = e^{\ln w}= w \)

\( T(\ln w) = h ( w) * \ln w \) so

\( e^{h(w)*\ln w*e^{-(h(w)*\ln w) }} = w \)

Just an idea.

Ivars

Hi Ivars -

yes, "upping up" is always interesting: even if it doesn't work, it sharpens the contour of the subject. We know some thing not only by the "what it can" but also (and sometimes even more in depth) by the "what it cannot".
So, let's see to what you can proceed...

Cordially -

Gottfried