An Identity of Andrews and a New Method for the Riordan Array Proof of Combinatorial Identities

TL;DR

This paper introduces a simple Riordan array-based method to prove and extend combinatorial identities, including a Fibonacci-Pascal identity originally discovered by Andrews, connecting generating functions and hypergeometric functions.

Contribution

The paper presents a novel, straightforward Riordan array technique for proving combinatorial identities and extends it to derive new identities with connections to hypergeometric functions.

Findings

New simple proof of Andrews' Fibonacci-Pascal identity

Extension of the method to derive additional identities

Establishment of a link between Riordan array generating functions and hypergeometric functions

Abstract

We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, most of them rather involved or else relying on sophisticated number theoretical arguments. We present a new proof, quite simple and based on a Riordan array argument. The main point of the proof is the construction of a new Riordan array from a given Riordan array, by the elimination of elements. We extend the method and as an application we obtain other identities, some of which are new. An important feature of our construction is that it establishes a nice connection between the generating function of the $A-$sequence of a certain class of Riordan arrays and hypergeometric functions.