02/12/2015, 10:27 AM

Hi.

I was wondering about this. I saw this:

http://arxiv.org/abs/1105.4735

made by two forum members here. It gives asymptotic formulae for the tetration and "upper super-exponential" to base , also called "".

In particular, the formula is given for the tetration at base with

so that is just a constant shift of this function. I'm not sure if this is a convergent formula or just an asymptotic, as the paper says asymptotic analysis determines the coefficients.

The first few polynomials are given by

Now, is there some kind of explicit (at least a finite summation/product etc. is what I mean) formula for these polynomials?

Thanks to Richard Stanley on mathoverflow.net here:

http://mathoverflow.net/questions/57627/...lso-how-ca

an explicit formula was found for the polynomial coefficients for the regular iteration at the base , in particular if

and

then

and for ,

where

are polynomials, and the coefficients are given explicitly by

where ranges over all subsets of such that . And

where is given for the given subset by ordering its elements in the order , as

.

where the -numbers are just the familiar Stirling second-kind numbers and .

So is there an explicit formula for the coefficients, or, perhaps more elegantly, the polynomials (e.g. as sums of multiplied simpler polynomials, for example) in the tetration? Also, are these formulas of any interest? I don't know if anyone saw them on the MathOverflow site. I'm the original poster of the question linked there, btw.

I was wondering about this. I saw this:

http://arxiv.org/abs/1105.4735

made by two forum members here. It gives asymptotic formulae for the tetration and "upper super-exponential" to base , also called "".

In particular, the formula is given for the tetration at base with

so that is just a constant shift of this function. I'm not sure if this is a convergent formula or just an asymptotic, as the paper says asymptotic analysis determines the coefficients.

The first few polynomials are given by

Now, is there some kind of explicit (at least a finite summation/product etc. is what I mean) formula for these polynomials?

Thanks to Richard Stanley on mathoverflow.net here:

http://mathoverflow.net/questions/57627/...lso-how-ca

an explicit formula was found for the polynomial coefficients for the regular iteration at the base , in particular if

and

then

and for ,

where

are polynomials, and the coefficients are given explicitly by

where ranges over all subsets of such that . And

where is given for the given subset by ordering its elements in the order , as

.

where the -numbers are just the familiar Stirling second-kind numbers and .

So is there an explicit formula for the coefficients, or, perhaps more elegantly, the polynomials (e.g. as sums of multiplied simpler polynomials, for example) in the tetration? Also, are these formulas of any interest? I don't know if anyone saw them on the MathOverflow site. I'm the original poster of the question linked there, btw.