I am searching for some transformations for a ${}_{3}F_2$ hypergeometric function which send the argument $z$ to $\frac{1}{1-z}$, or any other forms where I can interpret ${}_3F_2$ as a Laplace transform. I am aware of the transformation sending $z$ to $\frac{1}{z}$:
Transformations relating 3F2 at z with 3F2's at 1/z
Of course, if there is a general formula for ${}_{p+1}F_p$, that would be much appreciated.
I understand how to do this for ${}_2F_1$, but there are two steps in the proof (Lebedev's book), and it seems we do not have that for ${}_3F_2$.