# Generalized binomial coefficients

• Apr 28th 2009, 04:32 AM
NonCommAlg
Generalized binomial coefficients
Definition: For any $x \in \mathbb{R}$ and integer $k \geq 0$ we define $\binom{x}{0}=1$ and $\binom{x}{k}=\frac{x(x-1) \cdots (x-k+1)}{k!}, \ \ k \geq 1.$

Prove that for any integers $m,n,k,$ where $k \geq 0$ and $n \geq 1,$ we have: $\binom{\frac{m}{n}}{k} \in \mathbb{Z}[1/n].$