As part of my project I'm asked to evaluate the positivity of the following difference: $$\binom{l_1+t-1}{l_1}\binom{l_2+m+t-1}{l_2+m}\sum_{j=0}^{t}\binom{t-j+m}{m}\binom{j+l_1}{j}\binom{j+l_2}{j}-\binom{l_1+t}{l_1}\binom{l_2+m+t}{l_2+m}\sum_{j=0}^{t-1}\binom{t-j+m-1}{m}\binom{j+l_1}{j}\binom{j+l_2}{j}$$
$l_1, l_2, m, t$ are all positive integers. The conjecture is that the difference is negative for all t.
I have asked a similar question on whether the sum of product of these binomial coefficients has a closed form expression. People told me there is not using WZ method. So factoring definitely does not work. I'm wondering if there's a general methods, or special observations, to tackle this sort of expression.
