Let $m,n,p$ positive integers with $m\geq n$ and $H_m=1+1/2+1/3+\cdots+1/m$ the $m-$ith Harmonic Number with $H_0:=0$.
Show that for the values of $m,p,n$ for which the denominators do not vanish, the following identity holds:
$\displaystyle{\begin{aligned}\sum_{k=0}^{n}(-1)^{k-1}\frac{\dbinom{n}{k}\dbinom{p}{m-k}(H_p+H_m-H_{m-k})}{(m+n-k)\dbinom{p+n}{m+n-k}}&=\sum_{k=0}^{n}(-1)^{k-1}\frac{\dbinom{n}{k}\dbinom{p+n-k}{m-k}(H_{p+n-k}+H_m-H_{m-k})}{(m+n-k)\dbinom{p+n-k}{m+n-k}}\notag \\ &=\frac{1}{nm\dbinom{p+n}{n}}\end{aligned}}$
Omran Kouba, Anastasios Kotronis