 Solution

$J$ is entire in $z$. Along the fixed Pochhammer contour, the continued logarithms of $t$ and $1-t$ are bounded, and the contour stays away from both branch points. The integrand $\exp ((z-1)\log t+(b-1)\log (1-t))$ is entire in $z$ with uniform bounds on each compact parameter set. Differentiation under the contour integral is therefore valid to every order.