SCHWEITZER 1997 Problem 3

Denote $f_n(X) \in \Bbb Z [X]$ the polynomial $\Pi_{j=1}^n ( X + j -1)$ . Show that if the numbers $\alpha$ and $\beta$ satisfy $f'_{1997} (\alpha) = f'_{1999} (\beta) = 0$ , then $f_{1997} (\alpha ) \neq f_{1999} (\beta)$ .