For non-negative integers s and r, let

$$\left( {\matrix{ s \cr r \cr } } \right) = \left\{ {\matrix{ {{{s!} \over {r!(s - r)!}}} & {if\,r \le \,s,} \cr 0 & {if\,r\, > \,s} \cr } } \right.$$

For positive integers m and n, let

$$g(m,\,n) = \sum\limits_{p = 0}^{m + n} {{{f(m,n,p)} \over {\left( {\matrix{ {n + p} \cr p \cr } } \right)}}} $$

where for any non-negative integer p,

$$f(m,n,p) = \sum\limits_{i = 0}^p {\left( {\matrix{ m \cr i \cr } } \right)\left( {\matrix{ {n + i} \cr p \cr } } \right)\left( {\matrix{ {p + n} \cr {p - i} \cr } } \right)} $$

Then which of the following statements is/are TRUE?