Yep, found it.
The function \( f(x) \) defined as series of n times iterated exponential \( \exp^{\circ n}_e(x) \) (with base e) terms divided by corresponding n-th hyperfactorials (2^3^4^...n....) is on p. 326-327, Berndt, Ramanujan Notebooks, Vol 1., in excerpts from Second Quaterly report. It would be interesting to see if the original contains something more.
Berndt states that it converges for every x and for every n>0 \( f(x)> \exp^{\circ n}_e(x) \).
He does not mention entire function there.
Ivars
P.S. by adhering to Berndt's endnote of giving way to formal math, we can define by simple analogy and make conjectures:
tetra e = 1+1/2^3+1/2^3^4+ .......... = 1+1/8 + rest = probably around 1,125...because of extreme slowness.
and use it as basis for taking tetra-logarithms of iterated exponential on iteration parameter ( if we have e[4] n = f(e,4,n) then basis tetra e would be used to take tetra-log f(e,4,n) = tetra-log (n). For other bases a the value would have to be adjusted. For other n-tations we need different base for penta log etc.
Further analogies may include tetra Taylor expansion, etc.
The function \( f(x) \) defined as series of n times iterated exponential \( \exp^{\circ n}_e(x) \) (with base e) terms divided by corresponding n-th hyperfactorials (2^3^4^...n....) is on p. 326-327, Berndt, Ramanujan Notebooks, Vol 1., in excerpts from Second Quaterly report. It would be interesting to see if the original contains something more.
Berndt states that it converges for every x and for every n>0 \( f(x)> \exp^{\circ n}_e(x) \).
He does not mention entire function there.
Ivars
P.S. by adhering to Berndt's endnote of giving way to formal math, we can define by simple analogy and make conjectures:
tetra e = 1+1/2^3+1/2^3^4+ .......... = 1+1/8 + rest = probably around 1,125...because of extreme slowness.
and use it as basis for taking tetra-logarithms of iterated exponential on iteration parameter ( if we have e[4] n = f(e,4,n) then basis tetra e would be used to take tetra-log f(e,4,n) = tetra-log (n). For other bases a the value would have to be adjusted. For other n-tations we need different base for penta log etc.
Further analogies may include tetra Taylor expansion, etc.