I have obtained a Mellin transform
$ \mathcal{M}[f](s)= \frac{ \Gamma\left( 1-\frac{2-s}{a}\right) \Gamma\left( \frac{2-s}{a}\right) \Gamma\left( \frac{s}{2}\right) {{2}^{s-1}}}{a \Gamma\left( 1-\frac{s}{2}\right) } =F(s) $ which is a kernel of a Fox H-function by definition. I am trying to identify $f (t)= \mathcal{M}^{-1}[F](t)$ but am not very familiar with the notation.
My guess is
$ H^{2,1}_{0,1} \left( \begin{array}{l} (2/a, 1/a), (2/a, -1/a), (0,1/2) \\ (1, -1/2) \end{array} \right) $
Can someone verify?