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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.

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    $\begingroup$ Whoa. This is really interesting. I'll post if I come up with anything. $\endgroup$ – K.defaoite Aug 12 '20 at 14:33
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    $\begingroup$ Could you please provide the definition of fractional integral you are using? There appears to be many conventions on the Wikipedia page. $\endgroup$ – K.defaoite Aug 12 '20 at 14:42
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    $\begingroup$ Sort of. There's quite a jump between $$\log(f(x))=\lim_{h\to 0}\frac{f^h(x)-1}{h}$$ and then $$\log(\mathrm{D}_x)=\lim_{h\to 0}\frac{\mathrm{D}_x^h-1}{h}.$$ A lot of details are needed to make sure this new operator is consistent. Regarding the validity $\mathrm{D}^\alpha\mathrm{D}^\beta=\mathrm{D}^{\alpha+\beta}$, while this may not be true for the Riemann Liouville definition, I believe it is true for the Grunwald-Letnikov :). For eigenfunctions of fractional derivatives, see here $\endgroup$ – K.defaoite Aug 14 '20 at 17:56
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    $\begingroup$ I looked at $\frac{\partial}{\partial a}D^a_x(f(x))$ many years ago, finding for example that the values of $x$ for which $\frac{d}{da}(\frac{d^a}{dx^a}(x))=0$ at $a=0$ are $x=0$ and $x=e^{-\gamma}$, the latter constant appearing all over the place in this. $\endgroup$ – ruffle Sep 11 '20 at 3:29
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    $\begingroup$ Another way to generalise is to go $\frac{\partial}{\partial a} D^a_x(f(x))$, $\frac{\partial}{\partial b}\frac{\partial ^b}{\partial a^b}D^a_x(f(x))$, $\frac{\partial}{\partial c}\frac{\partial ^c}{\partial b^c}\frac{\partial ^b}{\partial a^b}D^a_x(f(x))$ and so on. $\endgroup$ – ruffle Sep 11 '20 at 3:38
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I've thought about this for a few days now, I didn't originally intend to answer my own question but it seems best to write this as an answer rather than add to the question. I think there is nice interpretation in the following: $$ f(x) = \lim_{h \to 0} \frac{e^{h f(x)}-1}{h} $$ also consider the Abel shift operator $$ e^{h D_x}f(x) = f(x+h) $$ from the limit form of the derivative we have (in the sense of an operator) $$ D_x = \lim_{h \to 0} \frac{e^{h D_x}-e^{0 D_x}}{h} = \lim_{h \to 0} \frac{e^{h D_x}-1}{h} $$ now we can also manipulate the first equation to get $$ \log f(x) = \lim_{h \to 0} \frac{f^h(x)-1}{h} $$ so by (a very fuzzy) extrapolation, we might have $$ \log(D_x) = \lim_{h \to 0} \frac{D_x^h-1}{h} $$ and applying that to a function we now get $$ \log(D_x) f(x) = \lim_{h \to 0} \frac{D_x^h f(x)-f(x)}{h} $$ which is the $\alpha = 0$ case of the 'super-derivative'. So one interpretation of this case is the logarithm of the derivative? If we apply the log-derivative to a fractional derivative then we have $$ \log(D_x) D^\alpha_x f(x) = \lim_{h \to 0} \frac{D_x^h D^\alpha_x f(x)-D^\alpha_x f(x)}{h} $$ there might be a question of the validity of $D_x^h D^\alpha_x = D_x^{\alpha+h}$ which I believe isn't always true for fractional derivatives.

This interpretation would explain the $\log(x)$ type terms arising in the series above. I'd be interested to see if anyone has any comments on this? I'd love to see other similar interpretations or developments on this. What are the eigenfunctions for the $\log D_x$ operator for example? Can we form meaningful differential equations?

Edit: For some functions I have tried we do have the expected property $$ n \log(D_x) f(x) = \log(D_x^n) f(x) $$ with $$ \log(D_x^n) f(x) = \lim_{h \to 0} \frac{D_x^{n h} f(x)-f(x)}{h} $$

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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.

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    $\begingroup$ Khesin et al. have written about the log of a derivative op, they but they consider pseudo differential ops, which are related to the Fourier transform rather than the Mellin and Laplace transform and the fractional derivatives (axiomatized by Pincherle) that you (and I) are interested in. $\endgroup$ – Tom Copeland Aug 28 '20 at 2:33
  • $\begingroup$ Thank you for these links, it will take some time to understand this but it looks promising. $\endgroup$ – Benedict W. J. Irwin Sep 15 '20 at 13:05

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