Log-Concavity and Log-Convexity of Restricted Infinite Products

In this paper we provide a classification on the sign distribution of Δ₄, (n): = p₄, (n) ² - p₄, (n-1) \, p₄, (n+1), where equation* ₍ =₀^ p₄, (n) \, qⁿ: = ₍ ₒ (1 - qⁿ) ^-f_{ (n) }, (N, f₁ 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, p₄, ₁ (n) represents the number of k-regular partitions. More generally, let f_ satisfy a certain growth condition. We determine the signs of Δ₄, (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* Δ₄, (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* Δ₄, (n) 4.