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

• Correct answer
  A

  $g\left(m, n\right)=g\left(n, m\right)$ for all positive integers $m, n$
• Correct answer
  B

  $g\left(m, n+1\right)=g\left(m+1,n\right)$ for all positive integers $m, n$
• C

  $g\left(2m, 2n\right)=2 g\left(m, n\right)$ for all positive integers $m, n$
• Correct answer
  D

  $g\left(2m, 2n\right)={\left(g\left(m,n\right)\right)}^2$ for all positive intergers $m, n$

Hence, the correct answer is options: 'A', 'B', 'D'.