Let g(z) be an interpolation of tetration that is entire.
The idea is to “add “ the log branches “ manually “ and at the same time “add “ the functional equation.
For instance let for some positive or zero x
\( g(z) = \sum_{n=0}^{oo} exp((z-n) exp^{[n]}(x)) exp^{[n]}(x) (-1)^n sin(\pi z) (\pi(z-n))^{-1} \)
This is an interpolation of tetration that is entire and fits perfectly at the values exp^[n](x) = g(n).
Now let the functional inverse of g(z) be f0(z).
This f0 now approximates an slog by interpolation.
Notice all this can be generalized to other bases or even other functions or sequences!
Now we try to correct this f0.
For x> 1
\( f_{n+1}(x) = 1+ f_n(ln(x)) \)
And then take the lim f_n= f_oo = S(z).
Now we take the functional inverse of S(z).
We call the inverse tet(z).
This tet(z) should be tetration.
I assume all this converges.
But I would like a formal proof that the lim f_oo exists.
This somewhat resembles the 2sinh method or the base change.
I assume since we started with entire we end up with C^oo functions at least.
We wonder if this is analytic.
And what criterion’s it satisfies ?
I repeat this can be used for other sequences too such as fibonacci like for instance.
Or iterations of polynomials.
Also other interpolations can be considered.
Variants are welcome.
Regards
Tommy1729
The idea is to “add “ the log branches “ manually “ and at the same time “add “ the functional equation.
For instance let for some positive or zero x
\( g(z) = \sum_{n=0}^{oo} exp((z-n) exp^{[n]}(x)) exp^{[n]}(x) (-1)^n sin(\pi z) (\pi(z-n))^{-1} \)
This is an interpolation of tetration that is entire and fits perfectly at the values exp^[n](x) = g(n).
Now let the functional inverse of g(z) be f0(z).
This f0 now approximates an slog by interpolation.
Notice all this can be generalized to other bases or even other functions or sequences!
Now we try to correct this f0.
For x> 1
\( f_{n+1}(x) = 1+ f_n(ln(x)) \)
And then take the lim f_n= f_oo = S(z).
Now we take the functional inverse of S(z).
We call the inverse tet(z).
This tet(z) should be tetration.
I assume all this converges.
But I would like a formal proof that the lim f_oo exists.
This somewhat resembles the 2sinh method or the base change.
I assume since we started with entire we end up with C^oo functions at least.
We wonder if this is analytic.
And what criterion’s it satisfies ?
I repeat this can be used for other sequences too such as fibonacci like for instance.
Or iterations of polynomials.
Also other interpolations can be considered.
Variants are welcome.
Regards
Tommy1729
