Turan inequalities for infinite product generating functions

In the 1970s, Nicolas proved that the partition function p(n) is log-concave for n > 25. In Heim et al. (Ann Comb 27(1):87-108, 2023), a precise conjecture on the logconcavity for the plane partition function pp(n) for n > 11 was stated. This was recently proven by Ono, Pujahari, and Rolen. In this paper, we provide a general picture. We associate to double sequences {gd (n)} d,n with gd (1) = 1 and 0 <== gd (n) - n(d) <= g(1) (n) (n - 1)(d-1), polynomials {P-n(gd) (x)} d,n given by Sigma n=0 infinity P-n(gd) ( x) qn := exp (x Sigma n=1 gd (n) qn n = 8 n=1 1 - qn -x fd (n). We recover p(n) = Ps1 n (1) and pp (n) = Ps2 n (1), where sd (n) := |n d and fd (n) = nd-1. Let n = 6. Then the sequence {P sd n (1)} d is log-concave for almost all d if and only if n is divisible by 3. Let id(n) = n. Then Pid n (x) = xn L (1) n-1(-x), where L (a) n (x) denotes the a-associated Laguerre polynomial. In this paper, we investigate Turan inequalities gd n (x) := P gd n ( x)2 - P gd n-1( x) P gd n+1(x) = 0. Let n = 6 and 0 = x < 2- 12 n+4. Then n is divisible by 3 if and only if gd n (x) = 0 for almost all d. Let n = 6 and n = 2 (mod 3). Then the condition on x can be reduced to x = 0. We determine explicit bounds. As an analogue to Nicolas' result, we have for g1 = id that id n (x) = 0 for all x = 0 and all n.