On the $\textrm {mod} 2$ reciprocation of infinite modular-part products and the parity of certain partition functions

Abstract:

An infinite, modular-part (MP) product is defined to be a product of the form ${\Pi _{n \in S}}(1 - {x^n})$, where $S = \{ {n \in {{\mathbf {Z}}^ + }:n \equiv {r_1}, \ldots ,{r_t}\;\pmod m} \}$. Some products of this kind have a $\bmod 2$ reciprocal that is also an MP product, while others do not. A complete method is first developed which determines if a given MP product has an MP reciprocal modulo 2 and finds it if it does. Next, a graph-theoretic interpretation of this method is made from which a streamlined algorithm is derived for deciding whether the given MP product is such a reciprocal. This algorithm is then applied to the single-variable Jacobi triple product and the quintuple product to determine the cases when these products have an MP reciprocal $\pmod 2$. When this occurs—and this occurs in infinitely many cases—the parity of the associated partition function can readily be found. A discussion is also made of the probability that a given MP product with modulus m has an MP reciprocal $\pmod 2$.