Problem

Source: China TST 2000, problem 2

Tags: induction, algebra, polynomial, factorial, combinatorics, Finite Differences



Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate \[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]