Question: Is there a closed form for the summation $$\sum_{k=0}^p \binom{k+m-1}{m-1}?$$
By writing out the summation, we have $$\binom{m-1}{m-1} + \binom{m}{m-1} + \binom{m+1}{m-1}+...+\binom{p+m-2}{m-1}+\binom{p+m-1}{m-1}$$ $$ = 1 + m + \frac{m(m+1)}{2} + \frac{m(m+1)(m+2)}{3!}+...+\frac{m(m+1)...(m+(k-1))}{k!}+...+\frac{m(m+1)...(m+(p-1))}{p!}.$$ However, I do not see a closed form for it.
In the summation $$\sum_{k=0}^p \binom{k+m-1}{m-1},$$ make a change of variable by letting $i = k + m - 1$. When $k = 0$, $i = m-1$; when $k = p$, $i = p+m-1$. The required summation becomes $$\sum_{i=m-1}^{p+m-1} \binom{i}{m-1}.$$ Apply the Hockey-stick identity $${\sum _{i=r}^{n}{i \choose r}={n+1 \choose r+1}}$$ with $n = p+m-1$ and $r = m - 1$, so that $$\sum_{k=0}^p \binom{k+m-1}{m-1} = \sum_{i=m-1}^{p+m-1} \binom{i}{m-1} = {p+m \choose m}.$$