I am trying to understand the fractional derivation and fractional integration. I found a representation of the fractional derivation operator in term of shift operator that is:
$D^n=(1-S)^n = \sum_{k=0}^{+\infty}\frac{\Gamma(k-n)S^k}{\Gamma(-k)\Gamma(n+1)}$
If I define the integration as the inverse of derivation, I must have
$I\cdot D = 1$
where I define $I$ as the integration operator.
So, my question is can I have a representation of the fractional integration in terms of operator $S$ as
$I^n\cdot D^n = D^{-n}\cdot D^n = 1$
namely
$I^n = \sum_{k=0}^{+\infty}\frac{\Gamma(k+n)S^k}{\Gamma(-k)\Gamma(-n+1)}$ ?