I am looking for a way to simplify this expression:
$$
\sum_{i=0}^{n-k-1} \sum_{j=0}^{k-1} \left[ {n-k-1 \choose i} {k-1 \choose j} ((-1)^{k-1-j} - (-1)^{n-k-1-i}) \times {(n+0.5)! \over (n-i-j)(n-i-j-1)} \times \left( \sum_{m=0}^{U-1} {(U-1)! ((n-i-j)/2)! \over (U-1-m)! ((n-i-j)/2+m)!}\times S^m \right) \right]
$$
where
$$ U = {n+i+j+1 \over 2}, $$
$$ k \in \mathbb{Z}, \mbox{ }k \ge 1,$$
$$ n \in \mathbb{Z}, \mbox{ }n \ge 2, \mbox{ }n \ge k,$$
$$S \ge 0.$$