For non-negative integers s and r, let
$ \binom{s}{r}= \begin{cases}\frac{s!}{r!(s-r)!} & \text { if } r \leq s \\ 0 & \text { if } r>s\end{cases} $
For positive integers m and n, let
$ (m, n) \sum_{\mathrm{p}=0}^{\mathrm{m}+\mathrm{n}} \frac{\mathrm{f}(\mathrm{~m}, \mathrm{n}, \mathrm{p})}{\binom{\mathrm{n}+\mathrm{p}}{\mathrm{p}}} $ where for any nonnegative integer p,
$ f(m, n, p)=\sum_{i=0}^{\mathrm{p}}\binom{m}{i}\binom{n+i}{p}\binom{p+n}{p-i} $
Then which of the following statements is/are TRUE?