Solution
= Solution
The meromorphic continuations of $F$ and the <gamma function> have the same recurrence and the same value at $1$. Hence their residues at every nonpositive integer are the same. Their difference $f=F-\Gamma$ therefore has removable singularities there and is analytic elsewhere, so it extends to an <entire function>. Equivalently, $f(z+1)=zf(z)$ and $f(1)=0$ show inductively that the apparent quotient singularities are removable.