 Solution

Take a Hankel contour $C_{\eta }$ around the positive real axis, going from infinity on its lower side toward a circle of radius $0<\eta <1$, clockwise around that circle, then out along the upper side. Use $0\le \arg z\le 2\pi$ and defineThis is an entire function of $t$: the small circle stays away from zero and the pole at $-1$, while the exponentially decaying rays converge uniformly for $t$ in compact subsets. Contour deformation shows independence of $\eta$. For $\Re t>-1$, let $\eta \to 0$. The upper ray contributes $F(t)$, the lower ray contributes $-e^{2\pi it}F(t)$, and the small-circle contribution vanishes. Thus the analytic continuation isThe only apparent singularities are at integers. At the nonnegative integers they are removable because the original integral is holomorphic throughout $\Re t>-1$. Hence singularities can occur only at $-1,-2,\dots$.

In fact all of those points are simple poles. At $t=-n-1$ the two rays cancel and the clockwise circle gives $H(-n-1)=-2\pi i{c}_n$, whereSince the denominator has derivative $-2\pi i$ at each integer, the residue at $-n-1$ is $c_n$. The parameter has been called $t$ consistently; the final use of $z$ for this parameter in the printed question is harmless notation.