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This post is motivated by observing that certain operators have "fractional analogs".

For example, let $f: R\to R$ be a differentiable function.
The differential operator $D^{1}(f)$ produces a new function.
For positive integers $n\in N$, we can define the nth derivative: $D^{n}(f)\equiv D^{1}\circ D^{n-1}(f)$.
The half-derivative is defined to be the function $D^{\frac{1}{2}}(f)\equiv H(f)$ where $H\circ H(f)=D^{1}(f)$.
More generally, the qth derivative is defined $D^{q}(f)$ for $q \geq 0$.
Conveniently, this fractional derivative definition agrees w/ the usual definition when $q\in N$.

The goal of this post is to try to think about fractional operators more generally. That is many operator definitions $F^{n}(f)$ work for positive integers, but when are they extended?

\begin{array}{|l|l|l|l|} \hline \text{Operation} & \text{Discrete Index set} & \text{Continuum Index set} & \text{Note} \\ \hline % +(a,b) & \sum_{i \in I} f(i) & \int_{i \in I} f(i) di & \\ \hline % *(a,b) & \prod_{i \in I} f(i) & \exp\left( \int_{i \in I} \ln f(i) di \right) & \text{If A is a matrix } A^{1/2}\equiv\{M:M^2=A\} \\ \hline % & n!\equiv \underset{i\in{1,\dots,n}}{\prod i} & x!\equiv \Gamma(x+1) & \Gamma(n+1)=n! \text{ when n is a positive integer} \\ \hline % \circ(f,g) & f^{n}(x)\text{ , } n \in N & f^{a}(x)\text{ , } a \in R & f^{1/2}(x)\equiv\{g:g^2(x)=f(x)\} \\ \hline % D^{1}(f) & D^{n}(f)\text{ , } n \in N & D^{a}(f)\text{ , } a \in R_+ & D^{1/2}(f)\equiv\{g:D^2(g)=f\} \\ \hline % \end{array}

What are other known operations which can be extended/generalized in this way?

Albert Zevelev
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    Fractional could mean things besides $\mathbb R$. For instance the Skolem-Mahler-Lech theorem can be proven by interpolating function iteration to include $p$-adic numbers. –  Sep 05 '20 at 20:49
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    Integration is not a fractional operation for summation. Your $\sum_{i \in I} f(i)$ and $\int_{i \in I} f(i) di$ are not in general equal for integer $i$. But see Euler-Maclaurin summation for a relation between them. – Jair Taylor Sep 05 '20 at 21:35

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Let $f : X \rightarrow X$ be an endofunction. Modulo issues of existence and uniqueness:

  • $f^{-n}$ can be defined as the function such that $f^{-n} \circ f^n = f^n \circ f^{-n} = \mathrm{id}_X$.

  • $f^{1/n}$ can be defined as the function such that $(f^{1/n})^n = f$.

  • Let $a$ be a sequence of rationals. $f^{\lim_n a_n}$ can be defined as $\lim_n f^{a_n}$.

We can also try to find a function $g : \mathbb{R} \rightarrow X \rightarrow X$ such that $g(n) = f^n$ for all $n \in \mathbb{N}$ while satisfying e.g. asymptotic properties (see the case of the gamma function) and properties like analyticity (see the example of extending tetration to complex heights).

See also fractional iteration and fractional dynamics for a general theory.

user76284
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