Hi everybody,

This is my first question so I hope I will correctly be following the rules!

I am looking for a simplification of the expression

$$ m! \sum_{k=0}^n \binom{n}{k} \binom{\alpha k}{m} x^k, $$ where $n,m$ are integers and $0 < \alpha < 1$. Is it a known generating function? Or does it exist a simpler expression?

In the same order I am also trying to simplify the expression

$$ \sum_{k=1}^n k! \binom{\alpha l}{k} S(n,k) x^k,$$

where $l$ is an integer?

Many thanks for your answers!