For non-negative integers s and r, let
\(\left( \begin{array}{c} s \\ r \end{array} \right) =\begin{cases} \frac{s!} {r!(s - r)!} & \quad \text{if } r\, ≤\,s\text{},\\ 0 & \quad \text{if } r\,>\,s \text{ } \end{cases}\)
For positive integers m and n, let
g(m,n) \( \displaystyle\sum_{p=0}^{m+n}\frac{f(m,n,p)}{\left(\! \begin{array}{c} n +p\\ p \end{array} \!\right)} \)
where for any nonnegative integer p,
f(m,n,p) = \( \displaystyle\sum_{i=0}^{p}{\left(\! \begin{array}{c} m\\ i \end{array} \!\right)} \left(\! \begin{array}{c} n+i\\ p \end{array} \!\right)\left(\! \begin{array}{c} p+n\\ p-i \end{array} \!\right)\)
Then which of the following statements is/are TRUE?
(A) (m,n) = g(n,m) for all positive integers m,n
(B) (m,n + 1) = g(m + 1,n) for all positive integers m,n
(C) (2m,2n) = 2g(m,n) for all positive integers m,n
(D) (2m,2n) = (g(m,n))2 for all positive integers m,n