I found that the following equation holds for integers $l$, $k$, and any $x \neq 0,1$,
$\tag{1} \sum\limits_{l = 0}^k {\left( { - 1} \right)^l } \left( {\begin{array}{*{20}c} k \\ l \\ \end{array}} \right)\frac{{x^{l} }}{{\left( {\frac{{1 + l}}{x} + k - l} \right)\left( {\frac{l}{x} + 1 + k - l} \right)}} = \frac{x\left( {1 - x} \right)^{k}}{{k + 1}} $
both in numerically by Matlab and analytically by Mathematica.
So I think there is a reference proving the equation. I have searched equaitons in Wolfram, Wiki, and some tables of series, But I couldn't find any related one.
Actually there were some equations looks like this $\tag{2} \sum\limits_{l = 0}^k {\left( { - 1} \right)^l } \left( {\begin{array}{*{20}c} k \\ l \\ \end{array}} \right)f(k,l,x) = g(k,x), $
but no help.
Also I tried in this way: break the equation into two terms like this
$\tag{3} \sum\limits_{l = 0}^k {\left( { - 1} \right)^l } \left( {\begin{array}{*{20}c} k \\ l \\ \end{array}} \right)\frac{{x^{l} }}{{\left( {\frac{{1 + l}}{x} + k - l} \right)}} + \sum\limits_{l = 0}^k {\left( { - 1} \right)^l } \left( {\begin{array}{*{20}c} k \\ l \\ \end{array}} \right)\frac{{x^{l} }}{{\left( {\frac{l}{x} + 1 + k - l} \right)}}, $
and ran it in Mathematica. But they result in Gauss hypergeometric functions, $_2 F_1 (-k,*,*,x)$, with some coefficients, respectively.
Also $\times2$, I tried to prove it by myself showing the equation holds for $k=0$ and any $x \neq0,1$, then it holds as well when $k+1$ by using the eq (1). but I couldn't...beacuse the $k+1$ case becomes a totally different equation compared to eq (1)...... lol
How can I prove it or find a proof?