<s>[INST] Can we express the following ordinary generating function? I wish to express the following power series 
$$ \sum_{k \ge 0} \binom{n-k}{m} x^k$$
where $n,m$ are positive integer such that  $0< m \le n$
 [/INST] $$
\begin{align}
\sum_{k=0}^\infty\binom{n-k}{m}x^k
&=\sum_{k=0}^n\binom{n-k}{m}x^k+\sum_{k=n+1}^\infty\binom{n-k}{m}x^k\\
&=\sum_{k=0}^n\binom{n-k}{m}x^k+\sum_{k=1}^\infty\binom{-k}{m}x^{k+n}\\
&=\sum_{k=0}^n\binom{n-k}{m}x^k+(-1)^m\sum_{k=1}^\infty\binom{m-1+k}{m}x^{k+n}\\
&=\sum_{k=0}^n\binom{n-k}{m}x^k+(-1)^m\sum_{k=1}^\infty\binom{m-1+k}{k-1}x^{k+n}\\
&=\sum_{k=0}^n\binom{n-k}{m}x^k+(-1)^mx^{n+1}\sum_{k=1}^\infty(-1)^{k-1}\binom{-m-1}{k-1}x^{k-1}\\
&=\underbrace{\sum_{k=0}^n\binom{n-k}{m}x^k}_{\text{polynomial}}-\frac{x^{n+1}}{(x-1)^{m+1}}
\end{align}
$$
If $n\lt m$, the polynomial part is $0$.
If $n\ge m$, the polynomial part has degree $n-m$:
$$
\begin{align}
\sum_{k=0}^n\binom{n-k}{m}x^k
&=\sum_{k=0}^{n-m}\binom{n-k}{m}x^k\\
&=\sum_{k=m}^n\binom{k}{m}x^{n-k}
\end{align}
$$
</s>