Bridging fractional iteration and fractional calculus

[tommy1729]

I thought I'd add here a relationship between "iterated matrices" and "iterated derivatives". This is a secondary thought to most of my work on fractional calculus; but it's masterful as a bridge between these operations. I am going to refer to this as Ramanujan; and "little circle method" mathematicians would look at it.

Let's let the Matrix \(A\) be a non-singular matrix; so that \(A^{-1}\) exists. To be simple; let's assume that \(A : \mathbb{C}^n \to \mathbb{C}^n\). And let's write that:

\[
e^{Ax} = \sum_{n=0}^\infty A^n \frac{x^n}{n!}\\
\]

We can safely assume that \(e^{Ax} : \mathbb{C}^n \to \mathbb{C}^n\). Where \(x\) produces a semi-group structure. From here, we can write:

\[
\frac{d^s}{dx^s} e^{Ax} = A^s e^{Ax}\\
\]

Where this is a linear operator applying \(\mathbb{C}^n \to \mathbb{C}^n\). We can set \(x=0\); and I'm just rewriting Ramanujan's master theorem as he wrote it:

\[
\Gamma(s) A^{-s} = \int_0^\infty e^{-Ax}x^{s-1}\,dx\\
\]

And we've fractionally iterated the matrix \(A: \mathbb{C}^n \to \mathbb{C}^n\). I avoided a lot of "singular moments" here. But if we can map the matrix \(A\) well enough; this discussion is entirely rigorous. It relates Daniel's work; Sheldon's work; Bo's work; Tommy's work; and all that matrix shit in a quantum physics./hilbert space shit.

Fractional calculus is just:

\[
\frac{d^s}{dA^s} : \mathbb{C}_{\Re(s) > 0} \times \mathcal{H} \to \mathcal{H}\\
\]

Where

\[
\mathcal{H} = \{ A : \mathbb{C}^n \to \mathbb{C}^n\,|\, A^{-1} :\mathbb{C}^n \to \mathbb{C}^n\}\\
\]

Then:

\[
\frac{d^s}{dA^s} e^{Ax} = x^s e^{Ax}
\]

And we can differentiate across \(A\) or \(x\); and still have the same rules.

This is the language I think in; and it's just a translation of much of the standard "tetration forum" language; and current literature.

In theory yeah.

But in practice ...

Differentiating a noninteger amount of times with respect to a nontrivial infinite square matrix that might not be diagonalizable ?!
That gives a non-unique infinite tensor with divergent norm ?! 

Or did you mean differentiating with respect to a vector ?

And that is just the last line of your answer.


regards

tommy1729