Une curieuse égalité entre deux sommes de produits de coefficients binomiaux

We will show in this text that, for all non-negative integers $n$ and $l$, the following equality is verified: $$\sum _{i=0}^l\left(\genfrac{}{}{0px}{}{n-i}{i}\right)\left(\genfrac{}{}{0px}{}{l+i}{2i+1}\right)=\sum _{i=0}^l\left(\genfrac{}{}{0px}{}{n-i}{i-1}\right)\left(\genfrac{}{}{0px}{}{l+i}{2i}\right)\mathrm{.}$$ We will first address the case where $l\le n$, for which both sums contain only classical binomial coefficients. Then, we will consider the general framework using generalized binomial coefficients.