Another question!

[mike3]

Let's take some function \( f(t) \) and some fractional differentiation method \( \frac{d^s}{dt^s} \) such that \( \frac{d^s f}{dt^s}(-t) < e^{-t} \)

Now create the function:

\( \phi(s) = \int_0^\infty t^{s-1} \frac{d^s f}{dt^s}(-t) dt \)

Integrate by parts, and for \( \Re(s) > 0 \) we get the spectacular identity that:

\( s \phi(s) = \phi(s+1) \)

What is going on here?

I'm not sure why this is necessarily bizarre. The functional equation you mention has infinitely many solutions. In general, \( \phi(x) = \Gamma(x) \theta(x) \) is a solution of \( x \phi(x) = \phi(x+1) \) for any 1-cyclic function \( \theta(x) \). If you take \( f(t) = e^t \) and use the Riemann-Liouville with lower bound \( -\infty \), then \( \frac{d^s f}{dt^s} = e^t \) and you recover the gamma function. I bet if you use another \( f \), you'll just get \( \Gamma(x) \theta(x) \) for some 1-cyclic function \( \theta(x) \) which is not just equal to 1.