Question

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!

Answer 1

Not sure that it helps, but letting $$ A_{m,n}(x) = m! \sum_{k=0}^n \binom{n}{k} \binom{\alpha k}{m} x^k $$ be the quantity you're studying, the generating function of $A_{m,n}(x)$ with respect to $m$ comes out to (if I haven't made an algebra error) $$ \sum_{m=0}^\infty A_{m,n}(x)\frac{y^m}{m!} = (1+x(1+y)^\alpha)^n. $$