General form for complex limit function $\sum p(n) z^n$ where $p \in \mathbb{C} [n]$

Question

Given a polynomial $p \in \mathbb{C} [n]$ of degree $k$, I need to show that the power series $\sum_{n=1}^{\infty} p(n) z^n$ uniformly converges in the open unit disc, and that the limit function $f$ is of the form $\frac{g(z)}{(1-z)^k}$ where $g$ is a polynomial of degree smaller than $k-1$.
As for the convergence question - I have set up upper and lower bounds for the radius using Cauchy-Hadamard theorem and showed that the radius is $1$. But I don't know how to show the rest.

Answer 1

Well, $$(1-z)f(z)=\sum_{n=0}^\infty p(n)(z^n-z^{n+1})=p(-1)+\sum_{n=0}^\infty(p(n)-p(n-1))z^n.$$ And $q(n)=p(n)-p(n-1)$ is a polynomial of degree $k-1$. Done by induction on $k$.