Log-Concavity and Log-Convexity of Restricted Infinite Products

In this paper we provide a classification on the sign distribution of ΔE,(n):= pE,(n)² - pE,(n-1) \, pE,(n+1), where {equation*} ∑n =0∞ pE,(n) \, q^n := ∏n ∈ S (1 - q^n )^{-f_{}(n)}, ( ∈ N, f_1≡ 1). {equation*} We take the product over 1∈ S ⊂ N and denote the complement by E, the set of exceptions. In the case of =1 and E the multiples of k, pE,1( n) represents the number of k-regular partitions. More generally, let f_ satisfy a certain growth condition. We determine the signs of ΔE,(n) for large. The signs mainly depend on the occurrence of subsets of \2,3,4,5\ as a part of the exception set and the residue class of n modulo r, where r depends on E. For example, let 2,3 ∈ S and $4$ an exception. Let n be large. Then for almost all we have {equation*} ΔE,(n) >0 \,\,\, for n≡ 2 {3}. {equation*} If we assume 3,4 ∈ S and $2$ an exception. Let n be large. Then for almost all we have {equation*} ΔE,(n) < 0 \,\,\, for n≡ 2 {3}. {equation*} Note that this property is independent of the integers k∈ S,k>4.