Hello everyone. So as I put in my last thread here, I have a solution to hyper operations for bases between 1 and \( \eta \). The natural question is if we can extend our results. As fivexthethird pointed out we can apply these ideas to the super root function. Then as Tommy pointed out this can then again result in a provable recursion.
To be clean and precise, I will explain the methodology here.
If \( ^s (srt(s,x)) = x \)
and if \( \frac{d^{s-1}}{dw^{s-1}}|_{w=0} \sum_{n=0}^\infty srt(n+1,x) \frac{w^n}{n!} \) converges and exists then quite exactly since \( srt(n+1,x) \) is the inverse of \( x^{x^{x^{...n+1\,times...}}} \) it is calcuable for \( x>\eta \) and then \( srt(s,x)=\frac{d^{s-1}}{dw^{s-1}}|_{w=0} \sum_{n=0}^\infty srt(n+1,x) \frac{w^n}{n!} \)
Inverting this function \( ssrt \) gives us a solution to tetration, proving the recursion with a little breath of thought. The problem is proving convergence of the differintegral. If we can do this we can iterate the procedure and solve for pentation then hexation and then septation, etc... This will give us a solution to hyper operations with a base on the positive real line. Not only that, an incredibly economical solution.
I've begun work on this and I think the procedure for proving convergence is the only problem, and then I think I got the knack of the result. I'm performing a similar induction schema and I think the result only needs a little bit of analysis magic that I've yet to find. But I'm sure it's out there.
I'm posting this thread to see if anyone has any ideas. I'm very excited about this result and it tittilates me to see it proved. I am wanting to collaborate on a paper with whomever wants to work on these ideas. I think some of the steps may be more complicated than first expected. I'm sort of starting with the idea of convergence, as I am sure that is the hardest hill to climb.
Thanks again, and hope you guys can help.
To be clean and precise, I will explain the methodology here.
If \( ^s (srt(s,x)) = x \)
and if \( \frac{d^{s-1}}{dw^{s-1}}|_{w=0} \sum_{n=0}^\infty srt(n+1,x) \frac{w^n}{n!} \) converges and exists then quite exactly since \( srt(n+1,x) \) is the inverse of \( x^{x^{x^{...n+1\,times...}}} \) it is calcuable for \( x>\eta \) and then \( srt(s,x)=\frac{d^{s-1}}{dw^{s-1}}|_{w=0} \sum_{n=0}^\infty srt(n+1,x) \frac{w^n}{n!} \)
Inverting this function \( ssrt \) gives us a solution to tetration, proving the recursion with a little breath of thought. The problem is proving convergence of the differintegral. If we can do this we can iterate the procedure and solve for pentation then hexation and then septation, etc... This will give us a solution to hyper operations with a base on the positive real line. Not only that, an incredibly economical solution.
I've begun work on this and I think the procedure for proving convergence is the only problem, and then I think I got the knack of the result. I'm performing a similar induction schema and I think the result only needs a little bit of analysis magic that I've yet to find. But I'm sure it's out there.
I'm posting this thread to see if anyone has any ideas. I'm very excited about this result and it tittilates me to see it proved. I am wanting to collaborate on a paper with whomever wants to work on these ideas. I think some of the steps may be more complicated than first expected. I'm sort of starting with the idea of convergence, as I am sure that is the hardest hill to climb.
Thanks again, and hope you guys can help.
