Une curieuse \'egalit\'e entre deux sommes de produits de coefficients binomiaux

TL;DR

This paper proves a surprising equality between two sums of products of binomial coefficients for all non-negative integers n and l, exploring both classical and generalized binomial coefficients.

Contribution

It establishes a novel binomial coefficient identity and extends the analysis to generalized binomial coefficients.

Findings

Proves a new binomial sum equality for all non-negative integers n and l.

Addresses both classical and generalized binomial coefficients.

Provides a framework for understanding binomial coefficient identities.

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 $l \leq n$, for which both sums contain only classical binomial coefficients. Then, we will consider the general framework using generalized binomial coefficients.