Hi everyone. I've been doing some exhaustive research into the Weyl differintegral and I've found a very beautiful approach to solving hyperoperators.
I'll be very frank. so I'll just state the following results:
If |_{w=0} = \phi(z) = \frac{1}{\Gamma(z)}\int_0^\infty f(-w)w^{z-1}\,dw)
Is the Weyl differintegral at zero, then I've shown some equivalencies. If
is kosher then it is defined for at least the left half plane ending at the line  = b)
and as well |\le C_\sigma e^{\alpha |y|}\,\,\,0 \le \alpha < \pi/2)
for all 
and |\le C_{y} e^{\rho|\sigma|})
for 
and for some 
Then:
 = \sum_{n=0}^\infty \phi(-n-z)\frac{w^n}{n!})
As Well we obtain a new expression for which is what I'm going for:
\Gamma(z) = \sum_{n=0}^\infty \phi(-n)\frac{(-1)^n}{n!(z+n)} + \int_1^\infty f(-w)w^{z-1}\,dw)
I am currently writing a paper on this. The results are solid. I've checked them very well. I can explain how I got to them, but they take a bit to write up.
So now we say, by the following.
Define the function:
}{x[z]y} = \sum_{n=0}^\infty \frac{(-1)^n}{(x[n]y)n!(n-z)} + \int_1^\infty\vartheta_x(y,-w)w^{-z-1}\,dw)
where
is an entire function of order zero defined by  = \sum_{n=0}^\infty \frac{w^n}{(x[n]y)n!})
Let us assume there exists an analytic continuation of natural hyperoperators (natural z) for natural values of x (in the first argument), and complex arguments in the second argument (in y). And If
and })
both are kosher
then 1/x[z+1]y is kosher
and
} {x[z+1](y+1)} = \sum_{n=0}^\infty \frac{(-1)^n}{(x[n+1]y+1)n!(n-z)} + \int_1^\infty (\sum_{k=0}^\infty \frac{(-w)^k}{(x[k+1](y+1))k!})w^{-z-1}\,dw)
But!!! Since}{x [z] (x[z+1]y)} = \sum_{n=0}^\infty \frac{(-1)^n}{(x[n](x[n+1]y)n!(n-z)} + \int_1^\infty (\sum_{k=0}^\infty\frac{(-w)^k}{(x[k](x[k+1]y))k!} w^{-z-1}\,dw)
We see we are only talking about the naturals and the results are EQUIVALENT!!!!!!!
Okay, I know. This is a big if. Essentially you need to prove that the following integral on the right converges for>0)
, that | < C_\sigma e^{\alpha|y|})
for 
when 
and that |\le C_y e^{\rho|\sigma|})
for some 
. Where 
is the following function
\omega(z,x,y) = \sum_{n=0}^\infty \frac{(-1)^n}{(x[n]y)n!(n-z)} + \int_1^\infty\sum_{k=0}^\infty \frac{(-w)^k}{(x[k]y)k!}w^{-z-1}\,dw)
If this happens and
is holomorphic in 
for all natural 
. THEN WE HAVE AN ANALYTIC CONTINUATION OF HYPER OPERATORS!!!!!!!!!!!!!
QUESTIONS COMMENTS!
I have the evaluation that:
I'll be very frank. so I'll just state the following results:
If
Is the Weyl differintegral at zero, then I've shown some equivalencies. If
Then:
As Well we obtain a new expression for
I am currently writing a paper on this. The results are solid. I've checked them very well. I can explain how I got to them, but they take a bit to write up.
So now we say, by the following.
Define the function:
where
Let us assume there exists an analytic continuation of natural hyperoperators (natural z) for natural values of x (in the first argument), and complex arguments in the second argument (in y). And If
then 1/x[z+1]y is kosher
and
But!!! Since
We see we are only talking about the naturals and the results are EQUIVALENT!!!!!!!
Okay, I know. This is a big if. Essentially you need to prove that the following integral on the right converges for
If this happens and
QUESTIONS COMMENTS!
I have the evaluation that: