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