Log-concavity of infinite product and infinite sum generating functions

Abstract

In this paper, we expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let [Formula: see text] be the double sequences [Formula: see text] or [Formula: see text]. We associate double sequences [Formula: see text] and [Formula: see text], defined as the coefficients of [Formula: see text] [Formula: see text] These coefficients are related to the number of partitions [Formula: see text], plane partitions [Formula: see text] of [Formula: see text], and Fibonacci numbers [Formula: see text]. Let [Formula: see text] and let [Formula: see text]. Then the coefficients are log-concave at [Formula: see text] for almost all [Formula: see text] in the exponential (involving [Formula: see text]) and geometric cases (involving [Formula: see text]). The coefficients are not log-concave for almost all [Formula: see text] in both cases, if [Formula: see text]. Let [Formula: see text]. Then the log-concave property flips for almost all [Formula: see text].