Generalization of an Identity of Andrews

TL;DR

This paper presents a simple proof of Andrews' Fibonacci identity, generalizes it to other arrays beyond Pascal's triangle, and derives new identities involving trinomial coefficients and Catalan's triangle.

Contribution

Provides an elementary proof of Andrews' Fibonacci identity and extends it to other combinatorial arrays, revealing new relationships with Fibonacci numbers.

Findings

Elementary proof of Andrews' identity

Generalization to other arrays beyond Pascal's triangle

New identities linking trinomial coefficients and Catalan's triangle to Fibonacci numbers

Abstract

We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, all of them rather involved or else relying on sophisticated number theoretical arguments. We not only give a simple and elementary proof, but also show the identity generalizes to arrays other than Pascal's triangle. As an application we obtain identities relating trinomial coefficients and Catalan's triangle to Fibonacci numbers.