Robbins and Ardila meet Berstel

07/29/2020

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by   Jeffrey Shallit, et al.

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In 1996, Neville Robbins proved the amazing fact that the coefficient of X^n in the Fibonacci infinite product ∏_n ≥ 2 (1-X^F_n) = (1-X)(1-X^2)(1-X^3)(1-X^5)(1-X^8) ⋯ = 1-X-X^2+X^4 + ⋯ 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.