An Euler-Like Product with Fibonacci Exponents

The pentagonal theorem for partitions is a consequence of the expansion of Euler’s famous product \( (1-y) (1-y^2) (1-y^3)(1-y^4)(1-y^5) \cdots \) $(1-y)(1-y^2)(1-y^3)(1-y^4)(1-y^5)\cdots$ We investigate the nature of the coefficients of the series expansion of \( (1-y) (1-y^2) (1-y^3)(1-y^5)(1-y^8) \cdots \) $(1-y)(1-y^2)(1-y^3)(1-y^5)(1-y^8)\cdots$ , 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 \(2\times 2\) $2\times 2$ matrices.