12/20/2007, 03:16 AM

Yes, super-roots are very interesting. In fact, I would really like to know if there is a recurrence equation for the coefficients of super-roots in general. I have done some initial research into series-expansions of super-roots, but I have not found any nice results yet. The only fruit of my research into this subject is the Taylor-Puiseux conversion that I found and talked about in Abel Functional Equation since it was useful to use it there. I use this term since a Puiseux series is any series involving logarithms, and that theorem allows one to convert between Taylor series and series involving logarithms in a very interesting way. Although it was much easier to prove than I thought at first, it is not trivial, since it confused me for the longest time.

As an example of the research that I did on super-roots, it is best to start with the research of Ioannis Galidakis, as he found (with the help of Leroy Quet) the first recurrence equation for the coefficients of integer tetration. Although he does not use the same terminology, Ioannis Galidakis uses a Puiseux series for tetration, for example, the Puiseux series for x^x^x is:

now, for comparison, the Taylor series for x^x^x is:

Now if we use Langrange series inversion to invert the first series, we will get "ln(x) = something", which is not what we want, so it is actually easier to use Lagrange series inversion to find the inverse of the second series to obtain:

and since the first series had a constant term of 1, this inversion is a Taylor series about 1! This is the first time I've ever seen anyone ever define super-roots (other than 2 and infinity) so I might have been the first to do this. But now that we can find the coefficients of super-roots through Lagrange inversion, what I am interested in is in a recurrence equation that allows us to calculate these coefficients much faster. Who knows what we'll find...

Andrew Robbins

As an example of the research that I did on super-roots, it is best to start with the research of Ioannis Galidakis, as he found (with the help of Leroy Quet) the first recurrence equation for the coefficients of integer tetration. Although he does not use the same terminology, Ioannis Galidakis uses a Puiseux series for tetration, for example, the Puiseux series for x^x^x is:

now, for comparison, the Taylor series for x^x^x is:

Now if we use Langrange series inversion to invert the first series, we will get "ln(x) = something", which is not what we want, so it is actually easier to use Lagrange series inversion to find the inverse of the second series to obtain:

and since the first series had a constant term of 1, this inversion is a Taylor series about 1! This is the first time I've ever seen anyone ever define super-roots (other than 2 and infinity) so I might have been the first to do this. But now that we can find the coefficients of super-roots through Lagrange inversion, what I am interested in is in a recurrence equation that allows us to calculate these coefficients much faster. Who knows what we'll find...

Andrew Robbins