TITLE: Closed expression for hypergeometric sum
QUESTION [6 upvotes]: I am trying to simplify an expression and find a closed form for
$$\sum_{m=0}^l \binom{s-m}{s-l} \binom{s-1+m}{s-1}x^m$$
How could I get rid of this summation?

REPLY [2 votes]: Mathematica says: 
$$\binom{s}{s-l} \, _2F_1(-l,s;-s;x).$$
(without hypergeometrics for special values of $x,$ like $x=1:$
$$\frac{(-1)^l (l-2 s-1)!}{l! (-2 s-1)!}$$