Turán inequalities for infinite product generating functions

The Ramanujan Journal, vol. 65, pp. 1849–1861

Abstract

Abstract In 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 =

Authors 2

1. Bernhard E. Heim corresponding Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University · University of Cologne

   Affiliation as printed

   Faculty of Mathematical and Natural Sciences, Mathematical Institute, University of Cologne, Weyertal 86–90, 50931, Cologne, Germany

   Lehrstuhl A für Mathematik, RWTH Aachen University, 52056, Aachen, Germany

2. Markus Neuhauser Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University · Kutaisi International University

   Affiliation as printed

   Kutaisi International University, 5/7, Youth Avenue, Kutaisi, 4600, Georgia

   Lehrstuhl A für Mathematik, RWTH Aachen University, 52056, Aachen, Germany