Some "Theorem" on the generalized superfunction

[JmsNxn]

If we find a proper definition of A+1 or (A^-1)+1 we can use binomial series to generalize its exponentiations
I wonder if this method works, and what about on other operators, like d/dx? I maybe test fractional calculus some day soon.



Regards, Leo

Hey, Leo.

So the fractional calculus approach is equivalent to binomial series. This was discussed a long time ago on here when I first pointed out the fractional calculus approach.

If I take,

\(
\phi^{\circ z} (\xi) = \frac{d^{z-1}}{dw^{z-1}}|_{w=0}\sum_{n=0}^\infty \phi^{\circ n+1}(\xi) \frac{w^n}{n!}\\
\)

This iteration will be equivalent to the expansion,

\(
\phi^{\circ z+1}(\xi) = \sum_{n=0}^\infty \binom{z}{n} \Delta^n[\phi]\\
\Delta^0[\phi] = \phi(\xi)\\
\Delta^1[\phi] = \phi^{\circ 2}(\xi) - \phi(\xi)\\
\Delta^{n}[\phi] = \sum_{j=0}^n \binom{n}{j}(-1)^{n-j} \phi^{\circ j+1}(\xi)\\
\)

And you can show if one converges, so does the other.

This Newton series will be equivalent to a binomial expansion.

This method will work in a lot of scenarios; the difficulty tends to be slow convergence.

Regards, James

I cannot believe I ever missed this application for Julia's equation. This is fantastic!

If \( \lambda \) is julia's function for \( f \); then,

\(
\int_0^s \lambda(z)\,dx\bullet z =f^{\circ s}(z)\\
\)