Log-concavity of infinite product and infinite sum generating functions.

In this paper, we expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let { g d (n) } d ≥ 0 , n ≥ 1 be the double sequences σ d (n) = ∑ ℓ | n ℓ d or ψ d (n) = n d . We associate double sequences { p g d (n) } and { q g d (n) } , defined as the coefficients of ∑ n = 0 ∞ p g d (n) t n : = ∏ n = 1 ∞ (1 − t n) − ∑ ℓ | n μ (ℓ) g d (n / ℓ) n , ∑ n = 0 ∞ q g d (n) t n : = 1 1 − ∑ n = 1 ∞ g d (n) t n . These coefficients are related to the number of partitions p (n) = p σ 1 (n) , plane partitions pp (n) = p σ 2 (n) of n , and Fibonacci numbers F 2 n = q ψ 1 (n). Let n ≥ 3 and let n ≡ 0 (mod 3). Then the coefficients are log-concave at n for almost all d in the exponential (involving p g d ) and geometric cases (involving q g d ). The coefficients are not log-concave for almost all d in both cases, if n ≡ 2 (mod 3). Let n ≡ 1 (mod 3). Then the log-concave property flips for almost all  d. [ABSTRACT FROM AUTHOR]