Robbins and Ardila meet Berstel

Abstract

In 1996, Neville Robbins proved the amazing fact that the coefficient of Xn in the Fibonacci infinite product∏n≥2(1−XFn)=(1−X)(1−X2)(1−X3)(1−X5)(1−X8)⋯=1−X−X2+X4+⋯ 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. I 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.