An Euler-Like Product with Fibonacci Exponents

Abstract The pentagonal theorem for partitions is a consequence of the expansion of Euler’s famous product (1-y) (1-y²) (1-y³) (1-y⁴) (1-y⁵) (1 - y) (1 - y 2) (1 - y 3) (1 - y 4) (1 - y 5) ⋯ We investigate the nature of the coefficients of the series expansion of (1-y) (1-y²) (1-y³) (1-y⁵) (1-y⁸) (1 - y) (1 - y 2) (1 - y 3) (1 - y 5) (1 - y 8) ⋯, in which the sequence of exponents is the Fibonacci numbers. As a part of the study of the combinatorial properties of the development of this product, we show that the series expansion coefficients are from \ -1, 0, 1\ - 1, 0, 1, and their behavior is determined by a monoid of twenty-five 2 2 2 × 2 matrices.