How can we find the sum of the following series
$$\sum_{i=0}^p \binom{m-q+1+i}{i} \binom{n+q-1-i}{n-i}=\sum_{i=0}^p\frac{(m-q+1-i)!}{ i! (m-q+1)!}\frac{ ( n + q-1-i)!}{ (q-1)! (n-i)!}$$ where $p < n,m$?
How can we find the sum of the following series
$$\sum_{i=0}^p \binom{m-q+1+i}{i} \binom{n+q-1-i}{n-i}=\sum_{i=0}^p\frac{(m-q+1-i)!}{ i! (m-q+1)!}\frac{ ( n + q-1-i)!}{ (q-1)! (n-i)!}$$ where $p < n,m$?