Robbins and Ardila meet Berstel

TL;DR

This paper demonstrates how computational methods can easily derive the Robbins-Ardila Fibonacci coefficient result from Berstel's transducer, highlighting the power of software-assisted proofs in combinatorics.

Contribution

It shows a straightforward computational approach to connect Berstel's transducer with Robbins and Ardila's Fibonacci coefficient theorem.

Findings

Computational techniques can derive Fibonacci coefficient properties from transducers.

Berstel's transducer simplifies the proof of Robbins-Ardila's result.

Software-based methods facilitate proofs in combinatorics.

Abstract

In 1996, Neville Robbins proved the amazing fact that the coefficient of $X^n$ in the Fibonacci infinite product $$ \prod_{n \geq 2} (1-X^{F_n}) = (1-X)(1-X^2)(1-X^3)(1-X^5)(1-X^8) \cdots = 1-X-X^2+X^4 + \cdots$$ is always either $-1$, $0$, or $1$. The same result was proved later by Federico Ardila using a different method. Meanwhile, in 2001, Jean Berstel gave a simple 4-state transducer that converts an "illegal" Fibonacci representation into a "legal" one. We show how to obtain the Robbins-Ardila result from Berstel's with almost no work at all, using purely computational techniques that can be performed by existing software.