Fox H-function transforms

Remarks

The Mellin transform of the kernel is $$\mathcal{M} H(s) = \frac{\prod_{j=1}^{m} \Gamma(b_j + B_j s) \prod_{j=1}^{n} \Gamma(1 - a_j - A_j s)}{\prod_{j=m+1}^{q} \Gamma(1 - b_j - B_j s) \prod_{j=n+1}^{p} \Gamma(a_j + A_j s)},$$ and formally $\mathcal{M} g(s) = \mathcal{M} H(s)\, \mathcal{M} f(1 - s)$.

Example: with all $A_j = B_j = 1$, the kernel is a Meijer G-function and the H-transform is a Meijer G-function transform. In particular $m = q = 1$, $n = p = 0$, $b_1 = 0$, $B_1 = 1$ gives the kernel $e^{-z}$, whose Mellin transform is $\Gamma(s)$, and the H-transform is the Laplace transform.

Example with a scale different from $1$: $m = q = 1$, $n = p = 0$, $b_1 = 0$, $B_1 = 1/2$ gives the kernel with Mellin transform $\Gamma(s/2)$, which is $2 e^{-z^2}$. The check is $\int_0^\infty 2 e^{-z^2} z^{s-1}\, dz = \Gamma(s/2)$, by the substitution $u = z^2$.

Non-example: if some $A_j$ or $B_j$ is $0$ or negative, the parameters are outside the definition of the H-function.