Tetration extension for bases between 1 and eta

[bo198214]

Upon closer study i think i have found a formula that actually works.

\( {}^ x b = \lim_{k\to \infty} \log_b ^k({}^ k b (\ln(b){}^ \infty b)^x-{}^ \infty b(\ln(b){}^ \infty b)^x+{}^ \infty b) \)

Also i have noticed that this can be more generalized to say,

if \( f(x)=b^x \)
then
\( f^n(x)= \lim_{k\to \infty} \log_b ^k( \exp_b^k(x) (\ln(b){}^ \infty b)^n-{}^ \infty b(\ln(b){}^ \infty b)^n+{}^ \infty b) \)

Actually you rediscovered the Kœnigs formula (2.24 in the (unfinished) overview paper).
\( f^{[w]}(z)=\lim_{k\to\infty}
f^{[-k]}\left((1-\lambda^{w})\cdot p+\lambda^{w}\cdot f^{[k]}(z)\right) \)
for \( f(x)=b^x \). \( \lambda \) is the derivative at the fixed point \( p={^\infty b} \), which is \( \lambda=\ln(p)=\ln(b^p)=p\ln(b)=\ln(b){^\infty b} \).

Good work!