Question

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?

• A. $g ( m , n )= g ( n , m )$ for all positive integers $m , n$
• B. $g ( m , n +1)= g ( m +1, n )$ for all positive integers $m , n$
• C. $g (2 m , 2 n )=2 g ( m , n )$ for all positive integers $m , n$
• D. $g (2 m , 2 n )=( g ( m , n ))^{2}$ for all positive integers $m , n$

Answer 1

Answer: (A), (B), (D)

$g ( m , n )= g ( n , m )$ for all positive integers $m , n$

Answer 2

(A) $g ( m , n )= g ( n , m )$ for all positive integers $m , n$
(B) $g ( m , n +1)= g ( m +1, n )$ for all positive integers $m , n$
(D) $g (2 m , 2 n )=( g ( m , n ))^{2}$ for all positive integers $m , n$