Turán inequalities for infinite product generating functions

Abstract

AbstractIn the 1970s, Nicolas proved that the partition function p(n) is log-concave for $$ n > 25$$ n > 25 . In Heim et al. (Ann Comb 27(1):87–108, 2023), a precise conjecture on the log-concavity for the plane partition function $${{\textrm{pp}}}(n)$$ pp ( n ) for $$n >11$$ 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 $$\{g_d(n)\}_{d,n}$$ { g d ( n ) } d , n with $$g_d(1)=1$$ g d ( 1 ) = 1 and $$\begin{aligned} 0 \le g_{d}\left( n\right) -n^{d}\le g_{1}\left( n\right) \left( n-1\right) ^{d-1}, \end{aligned}$$ 0 ≤ g d n - n d ≤ g 1 n n - 1 d - 1 , polynomials $$\{P_n^{g_d}(x)\}_{d,n}$$ { P n g d ( x ) } d , n given by $$\begin{aligned} \sum _{n=0}^{\infty } P_n^{g_d}(x) \, q^n := {{\textrm{exp}}}\left( x \sum _{n=1}^{\infty } g_d(n) \frac{q^n}{n} \right) =\prod _{n=1}^{\infty } \left( 1 - q^n \right) ^{-x f_d(n)}. \end{aligned}$$ ∑ n = 0 ∞ P n g d ( x ) q n : = exp x ∑ n = 1 ∞ g d ( n ) q n n = ∏ n = 1 ∞ 1 - q n - x f d ( n ) . We recover $$ p(n)= P_n^{\sigma _1}(1)$$ p ( n ) = P n σ 1 ( 1 ) and $${{\textrm{pp}}}\left( n\right) = P_n^{\sigma _2}(1)$$ pp n = P n σ 2 ( 1 ) , where $$\sigma _d (n):= \sum _{\ell \mid n} \ell ^d$$ σ d ( n ) : = ∑ ℓ ∣ n ℓ d and $$f_d(n)= n^{d-1}$$ f d ( n ) = n d - 1 . Let $$n \ge 6$$ n ≥ 6 . Then the sequence $$\{P_n^{\sigma _d}(1)\}_d$$ { P n σ d ( 1 ) } d is log-concave for almost all d if and only if n is divisible by 3. Let $${{\textrm{id}}}(n)=n$$ id ( n ) = n . Then $$P_n^{{{\textrm{id}}}}(x) = \frac{x}{n} L_{n-1}^{(1)}(-x)$$ P n id ( x ) = x n L n - 1 ( 1 ) ( - x ) , where $$L_{n}^{\left( \alpha \right) }\left( x\right) $$ L n α x denotes the $$\alpha $$ α -associated Laguerre polynomial. In this paper, we investigate Turán inequalities $$\begin{aligned} \Delta _{n}^{g_d}(x) := \left( P_n^{g_d}(x) \right) ^2 - P_{n-1}^{g_d}(x) \, P_{n+1}^{g_d}(x) \ge 0. \end{aligned}$$ Δ n g d ( x ) : = P n g d ( x ) 2 - P n - 1 g d ( x ) P n + 1 g d ( x ) ≥ 0 . Let $$n \ge 6$$ n ≥ 6 and $$0 \le x < 2 - \frac{12}{n+4}$$ 0 ≤ x < 2 - 12 n + 4 . Then n is divisible by 3 if and only if $$\Delta _{n}^{g_d}(x) \ge 0$$ Δ n g d ( x ) ≥ 0 for almost all d. Let $$n \ge 6$$ n ≥ 6 and $$n \not \equiv 2 \pmod {3}$$ n ≢ 2 ( mod 3 ) . Then the condition on x can be reduced to $$x \ge 0$$ x ≥ 0 . We determine explicit bounds. As an analogue to Nicolas’ result, we have for $$g_1= {{\textrm{id}}}$$ g 1 = id that $$\Delta _{n}^{{{\textrm{id}}}}(x) \ge 0$$ Δ n id ( x ) ≥ 0 for all $$x \ge 0 $$ x ≥ 0 and all n.