An Euler-Like Product with Fibonacci Exponents

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.