Question: Let $k$ be a positive interger number, and define the sequence $\{P\_k(n)\}\_n$ as: $P\_k(0)=0$, $P\_k(1)=1$ and for every $n \geq 2$, $P\_{k}(n)=k\cdot P\_k(n-1)+P\_k(n-2)$. I want to find: 

1-$H\_k=\lim\_{ n \to \infty }\frac{P\_k(n)}{P\_k({n-1})}$ and 

2-$M=\sum\_{k=1}^{\infty}({H\_k }-k)$

Answer: 
**Hint** If you assume by contradiction that $-1 \in U$, then prove first that $f(z)=g(z)$ on $U \backslash \{-1\}$.

Then use the fact that $f$ is continuous at $-1$, and hence 
$$f(-1)=\lim\_{z \to -1} \frac{z}{z+1}=\infty$$
Hint: Note that a holomorphic function on a open and connected set $ U $ of $\mathbb {C}$ is uniquely determined by a non discrete subset of $U$.