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I confess, I'm a bit of a dilettante with respect to mathematics; But one thing I've been interested in is generalizations of abstractions. So naturally when I heard about the possibility of getting fractional differential orders, such as the half derivative, I was delighted. However one thing I did notice is that perusing the traditional wiki's that discussed the topic, the extension of these orders were limited to the complex plane.

Now its fascinating that you can do analytic continuation of the factorial into the gamma function and then make definitions of fractional differintegrals that use the traditional Riemann–Liouville integral to generalize the concept of the derivative to the reals or complex numbers:

$$ f^{(q)}(x) = \frac{1}{\Gamma(k-q)} \frac{d^k}{dx^k} \int_{a}^{x}\, (x-t)^{k-q-1}\,f(t)\,dt\>, \quad (k-1 < q < k )\,, $$

And there have been several extensions to the orthodox definition for dealing with periodic functions and the like. But most of the generalizations, as far as I can tell, at the heart of them rely on the gamma function.

$$ \Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,. $$

And at the heart of that is the exponential integral.

As far as I can tell, you should be able to extend the concept of the differintegral operator to arbitrary algebras that support exponentiation, such as hypercomplex numbers and various matrices, but I haven't read about anyone doing that. Looking at the matrix exponential, it seems well defined.

If this approach is correct, can this be further generalized to Riesz potentials of arbitrary algebras?

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