Log-concavity and log-convexity of restricted infinite products

In this paper, we provide a classification of the sign distribution of [Formula: see text], where [Formula: see text] We take the product over [Formula: see text] and denote the complement by [Formula: see text], the set of exceptions. When [Formula: see text] and [Formula: see text] is the set of multiples of [Formula: see text], [Formula: see text] represents the number of [Formula: see text]-regular partitions. More generally, let [Formula: see text] satisfy a certain growth condition. We determine the signs of [Formula: see text] for [Formula: see text] large. The signs mainly depend on which of [Formula: see text] belong to the exception set and the residue class of [Formula: see text] modulo [Formula: see text], where [Formula: see text] depends on [Formula: see text]. For example, let [Formula: see text] and [Formula: see text] is an exception. Let [Formula: see text] be large. Then for almost all [Formula: see text] we have [Formula: see text] If we assume [Formula: see text] and [Formula: see text] is an exception. Let [Formula: see text] be large. Then for almost all [Formula: see text] we have [Formula: see text] Note that this property is independent of the integers [Formula: see text].