Constructing a family of polynomials in $\mathbb R_n[X]$ Let $\mathbb R_n[X]$ be the vector space of polynomials of degree at most $n$.
Let $u$ the endomorphism sending $P$ to $P(X+1)-P(X)$. I want to show that there exists a unique family of polynomials $Q_0, \cdots,Q_n$ such that $Q_0=1$ and $u(Q_{k})=Q_{k-1}$ and $Q_k(0)=0$ for $k=1..n$.
I know so far that the map sending $P$ to $P(X+1)$ is an automorphism and that $\ker(u)=\mathbb R_{0}[X]$ and $Im(u)=R_{n-1}[X]$ but don't know how to go further. Thank you for your help!

Existence: $Q_k(X) := \binom{X}{k} := \frac{X(X-1)\dotsc(X-k+1)}{k!}$ does the job.
Uniqueness: If $Q_0,\dotsc,Q_n$ satisfy the requirements, then since $\ker(u)=\mathbb{R}$ we easily get $Q_k = c \binom{X}{k}$ for some constant $c$. But then $Q_0=1$ gives $c=1$.
