I am trying to digest an old paper by Kermack & McCrea (see https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/on-professor-whittakers-solution-of-differential-equations-by-definite-integrals-part-i/51CD535BEDC419E6763C28641DF0709B )

Here and there, they use fractional derivatives with which I am not quite familiar. For example, they use the formula

$$ \left(\frac{d}{dt}\right)^{1/2} t = t \left(\frac{d}{dt}\right)^{1/2} + 1/2 \left(\frac{d}{dt} \right)^{-1/2} $$

which seems to be really a straightforward generalization of Leibniz's rule. In much the same vein, we should have for example

$$ \left(\frac{d}{dt}\right)^{-1} t = t \left(\frac{d}{dt}\right)^{-1} - \left(\frac{d}{dt} \right)^{-2} $$ and so on. Correct? Can these formal manipulation be made rigorous somehow? Any suggestions, references?


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