Partial Differential Equation for power-towers

[kobi_78]

Hi, I found a pde for power towers of any height.

The equation is as following:

Define
\( f_{-1}(a, x) = \log_a{x} \)
\( f_0(a, x) = x \)
\( f_{n + 1}(a, x) = a^{f_n(a, x)} \)

And

\( G(x, a) = \frac{1}{a \ln(a)^2 } \sum_{k = -1}^{\infty}{ \frac{1}{\frac{df_{k}}{dx}}} = \frac{1}{a (\ln(a))^2 } \sum_{k = -1}^{\infty}{ \frac{1}{ \prod_{n=1}^{k}{f_{n}(a, x)} \cdot (\ln{a})^k }} \)

Where
\( \prod_{n=1}^{0}{f_{n}(a, x)} = 1 \)
\( \prod_{n=1}^{-1}{f_{n}(a, x)} = \frac{1}{f_0(a, x)} \)

Then every \( y = f_n(a, x) \) satisfies:

\( \frac{dy}{da} = G(x, a) \cdot \frac{dy}{dx} - G(y, a) \)

I don't understand anything about pdes so I don't know if this says any thing, but I wanted to share it with you guys.

I guess we can somehow use this equation to extend natural iteration monomial for larger bases, but I don't really know.

Kobi

[mike3]

Hmm. Well if this is the case, then could that mean that there may be a continuum of solutions? If so, could that allow us to define \( f_h(a, x) \) for real and complex values of \( h \), thereby yielding \( \exp^h_a(x) \) for such heights?