On the Coefficients of a Fibonacci Power Series

We give an explicit description of the coefficients of the formal power series (1− x)(1− x)(1 − x)(1− x)(1 − x)(1− x) · · ·. In particular, we show that all the coefficients are equal to −1, 0 or 1. The Fibonacci numbers are defined by the recurrence relation Fn+2 = Fn+1+ Fn for n ≥ 0, and the initial conditions F0 = 0, F1 = 1. Consider the infinite product A(x) = ∏ k≥2(1− x k) = (1− x)(1 − x)(1− x)(1 − x)(1 − x) · · · = 1− x− x + x + x − x + x − x − x + x + x + · · · regarded as a formal power series. In [4], N. Robbins proved that the coefficients of A(x) are all equal to −1, 0 or 1. We shall give a short proof of this fact, and a very simple recursive description of the coefficients of A(x). Following the notation of [4], let a(m) be the coefficient of x in A(x). It is clear that a(m) = rE(m) − rO(m), where rE(m) is equal to the number of partitions of m into an even number of distinct positive Fibonacci numbers, and rO(m) is equal to the number of partitions of m into an odd number of distinct positive Fibonacci numbers. We call these partitions “even” and “odd” respectively. Proposition 1. Let n ≥ 5 be an integer. Consider the coefficients a(m) for m in the interval [Fn, Fn+1). Split this interval into the three subintervals [Fn, Fn + Fn−3 − 2], [Fn + Fn−3 − 1, Fn + Fn−2 − 1] and [Fn + Fn−2, Fn+1 − 1]. 1. The numbers a(Fn), a(Fn + 1), . . . , a(Fn + Fn−3 − 2) are equal to the numbers (−1)a(Fn−3−2), (−1)a(Fn−3−3), . . . , (−1)a(0) in that order. 2. The numbers a(Fn + Fn−3 − 1), a(Fn + Fn−3), . . . , a(Fn + Fn−2 − 1) are equal to 0. 3. The numbers a(Fn + Fn−2), a(Fn + Fn−2 + 1), . . . , a(Fn+1 − 1) are equal to the numbers a(0), a(1), . . . , a(Fn−3 − 1) in that order.