Question

For nonnegative integers $s$ and $r$, let
$\left(\begin{array}{c}s\\ r\end{array}\right)=\{\begin{array}{ll}\frac{s!}{r!(s-r)!}& \text{ if }r\le s\\ 0& \text{ if }r>s\end{array}$
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?

• 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. $g(2m,2n)=2g(m,n)$ for all positive integers $m,n$
• D. $g(2m,2n)=(g(m,n){)}^2$ for all positive integers $m,n$

Answer 1

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

$f(m,n,p)=\sum _{i=0}^p\left({}^m{C}_i{}^{n+i}{C}_p{}^{p+n}{C}_{p-i}\right)$
$=\sum _{i=0}^p\left(\frac{m!}{i!(m-1)!}\cdot \frac{(n+i)!}{p!(n+i-p)!}\cdot \frac{(p+n)!}{(p-i)!(n+i)!}\right)$
$=\sum _{i=0}^p{}^m{C}_i\cdot {}^{n+p}{C}_n\cdot {}^n{C}_{p-i}$
$={}^{n+p}{C}_n\cdot {}^{m+n}{C}_p$
$g\left(m_{1}n\right)=\sum _{p=0}^{m+n}\frac{{}^{m+n}{C}_p{}^{n+p}{C}_n}{{}^{n+p}{C}_p}=\sum _{p=0}^{m+n}{}^{m+n}{C}_p$
$=2^{m+n}.$