Robbins and Ardila meet Berstel

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.