Une curieuse égalité entre deux sommes de produits de coefficients binomiaux

Abstract

We will show in this text that, for all non-negative integers $n$ and $l$, the following equality is verified: \[\sum_{i=0}^{l} {n-i \choose i}{l+i \choose 2i+1}=\sum_{i=0}^{l} {n-i \choose i-1}{l+i \choose 2i}.\] 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.

La langue de ce document est le français, with translation to English