TITLE: Is there any meaning to this "Super Derivative" operation I invented?
QUESTION [33 upvotes]: Does anyone know anything about the following "super-derivative" operation? I just made this up so I don't know where to look, but it appears to have very meaningful properties. An answer to this question could be a reference and explanation, or known similar idea/name, or just any interesting properties or corollaries you can see from the definition here? Is there perhaps a better definition than the one I am using? What is your intuition for what the operator is doing (i.e. is it still in any sense a gradient)? Is there a way to separate the log part out, or remove it? Or is that an essential feature?
Definition: I'm using the word "super-derivative" but that is a made-up name. Define the "super-derivative", operator $S_x^{\alpha}$, about $\alpha$, using the derivative type limit equation on the fractional derivative operator $D_x^\alpha$
$$
S_x^{\alpha}  = \lim_{h \to 0} \frac{D^{\alpha+h}_x-D^{\alpha}_x}{h}
$$
then for a function
$$
S_x^{\alpha} f(x) = \lim_{h \to 0} \frac{D^{\alpha+h}_xf(x)-D^{\alpha}_x f(x)}{h}
$$
for example, the [Riemann-Liouville, see appendix] fractional derivative of a power function is
$$
D_x^\alpha x^k = \frac{\Gamma(k+1)}{\Gamma(k-\alpha+1)}x^{k-\alpha}
$$
and apparently
$$
S_x^{\alpha} x^k =  \frac{\Gamma (k+1) x^{k-\alpha} (\psi ^{(0)}(-\alpha+k+1) - \log (x))}{\Gamma (-\alpha+k+1)} = (\psi ^{(0)}(-\alpha+k+1) - \log (x)) D_x^\alpha x^k
$$
a nice example of this, the super-derivative of $x$ at $\alpha=1$ is $-\gamma - \log(x)$, which turns up commonly. I'm wondering if this could be used to describe the series expansions of certain functions that have log or $\gamma$ terms, e.g. BesselK functions, or the Gamma function.
Potential relation to Bessel functions: For example, a fundamental function with this kind of series, (the inverse Mellin transform of $\Gamma(s)^2$), is $2 K_0(2 \sqrt{x})$ with
$$
2 K_0(2 \sqrt{x}) = (-\log (x)-2 \gamma )+x (-\log (x)-2 \gamma +2)+\frac{1}{4} x^2 (-\log (x)-2 \gamma +3)+\\
+\frac{1}{108} x^3 (-3 \log (x)-6 \gamma +11)+\frac{x^4 (-6 \log (x)-12 \gamma +25)}{3456}+O\left(x^5\right)
$$
in the end, taking the super-derivative of polynomials and matching coefficients we find
$$
S_x^1[2 \sqrt{x}I_1(2\sqrt{x})] + I_0(2 \sqrt{x})\log(x) = 2K_0(2 \sqrt{x})
$$
which can also potentially be written in terms of linear operators as
$$
[2 S_x x D_x + \log(x)]I_0(2 \sqrt{x}) = 2K_0(2 \sqrt{x})
$$
likewise
$$
[2 S_x x D_x - \log(x)]J_0(2 \sqrt{x}) = \pi Y_0(2 \sqrt{x})
$$
I like this because it's similar to an eigensystem, but the eigenfunctions swap over.
Gamma Function: We can potentially define higher-order derivatives, for example
$$
(S_x^{\alpha})^2  = \lim_{h \to 0} \frac{D^{\alpha+h}_x-2 D^{\alpha}_x + D^{\alpha-h}_x}{h^2}
$$
and
$$
(S_x^{\alpha})^3 = \lim_{h \to 0} \frac{D^{\alpha+3h}_x-3 D^{\alpha+2h}_x + 3 D^{\alpha+h}_x - D^{\alpha}_x}{h^3}
$$
this would be needed if there was any hope of explaining the series
$$
\Gamma(x) =  \frac{1}{x}-\gamma +\frac{1}{12} \left(6 \gamma ^2+\pi ^2\right)
    x+\frac{1}{6} x^2 \left(-\gamma ^3-\frac{\gamma  \pi ^2}{2}+\psi
    ^{(2)}(1)\right)+
\\+\frac{1}{24} x^3 \left(\gamma ^4+\gamma ^2 \pi ^2+\frac{3
    \pi ^4}{20}-4 \gamma  \psi ^{(2)}(1)\right)+O\left(x^4\right)
$$
using the 'super-derivative'. This appears to be
$$
\Gamma(x) = [(S^1_x)^0 x]_{x=1} x^{-1} + [(S^1_x)^1 x]_{x=1} x +  \frac{1}{2}[(S^1_x)^2 x]_{x=1} x^2 + \frac{1}{6} [(S^1_x)^3 x]_{x=1} x^3 + \cdots
$$
so one could postulate
$$
\Gamma(x) = \frac{1}{x}\sum_{k=0}^\infty \frac{1}{k!}[(S^1_x)^k x]_{x=1} x^{k}
$$
which I think is quite beautiful.
Appendix: I used the following definition for the fractional derivative:
$$
D_x^\alpha f(x) = \frac{1}{\Gamma(-\alpha)}\int_0^x (x-t)^{-\alpha-1} f(t) \; dt
$$
implemented for example by the Wolfram Mathematica code found here
FractionalD[\[Alpha]_, f_, x_, opts___] := 
  Integrate[(x - t)^(-\[Alpha] - 1) (f /. x -> t), {t, 0, x}, 
    opts, GenerateConditions -> False]/Gamma[-\[Alpha]]

FractionalD[\[Alpha]_?Positive, f_, x_, opts___] :=  Module[
  {m = Ceiling[\[Alpha]]}, 
  If[\[Alpha] \[Element] Integers, 
    D[f, {x, \[Alpha]}], 
    D[FractionalD[-(m - \[Alpha]), f, x, opts], {x, m}]
  ]
]

I'm happy to hear more about other definitions for the fractional operators, and whether they are more suitable.

REPLY [3 votes]: Seems like you have happened upon some relations similar to ones I've written about over several years. Try for starters the MSE-Q&A "Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$." There are several posts on my blog (see my user page) on this topic, logarithm of the derivative operator (see also A238363 and links therein, a new one will be added soon, my latest blog post), and fractional differ-integral calculus.