I've asked this question in two ways, as I think either is a solution to my problem. Assume that I have some distribution $f(t)$ for which I know the Mellin Transform $M(s)$ and the Laplace transform $L(p)$.
I am trying to find "half moments", i.e. $k \in -3/2, -1/2, 1/2, \cdots$, (or arbitrary moments, if that's easier) of a shift in the distribution, i.e.
$$E(t^s f(t-a)) \equiv \int_a^\infty t^s f(t-a)dt$$
I'm not sure if there is a simple formula for a "shifted" Mellin transform (which would be exactly what I need) or whether there is a way to derive "half" or arbitrary moments from the shifted Laplace, i.e. $e^{-a} L(p)$, as I know we can derive "normal" moments, i.e. $s \in N$, by taking the derivative.