Anybody shell know this very important formula:
\( \operatorname{sexp}_b(x)=r+\sum _{n=1}^{\infty} \frac{\left(\ln b \right)^{n-1}\left(\ln \left(b^r\right)\right)^{n x}\left(1-r)^n B_n^x}{n!} \)
where \( B_n^x \) are the Bell's numbers of x-th order and \( r=\frac{W(-\log (b))}{\log (b)} \). For integer x one can find them here:
http://www.research.att.com/~njas/sequences/A111672
http://www.research.att.com/~njas/sequences/A144150
http://www.research.att.com/~njas/sequences/A153277
This formula can be easily derived from regular teration, but has a long history dating from 1945 ( J. Ginsburg, Iterated exponentials, Scripta Math. 11 (1945), 340-353.)
It is notable that tetration and Bell's polynomials of n-th order have applications in quantum physics: http://arxiv.org/abs/0812.4047
\( \operatorname{sexp}_b(x)=r+\sum _{n=1}^{\infty} \frac{\left(\ln b \right)^{n-1}\left(\ln \left(b^r\right)\right)^{n x}\left(1-r)^n B_n^x}{n!} \)
where \( B_n^x \) are the Bell's numbers of x-th order and \( r=\frac{W(-\log (b))}{\log (b)} \). For integer x one can find them here:
http://www.research.att.com/~njas/sequences/A111672
http://www.research.att.com/~njas/sequences/A144150
http://www.research.att.com/~njas/sequences/A153277
This formula can be easily derived from regular teration, but has a long history dating from 1945 ( J. Ginsburg, Iterated exponentials, Scripta Math. 11 (1945), 340-353.)
It is notable that tetration and Bell's polynomials of n-th order have applications in quantum physics: http://arxiv.org/abs/0812.4047
