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