Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 7D / a / Solution

For any $n$ with $\mathrm{Re}z>-n$, defineThe definitions agree on overlaps by the recurrence, giving a meromorphic continuation to $\mathbb{C}$. Its only singularities are simple poles at $0,-1,-2,\dots$. Since $F(1)=1$, repeated use of the recurrence givesThese residues are nonzero, so all the listed singularities really are simple poles.