Log-Concavity and Log-Convexity of Restricted Infinite Products
Abstract
In this paper we provide a classification on the sign distribution of $\Delta _{E,\ell}(n):= p_{E,\ell }(n)^2 - p_{E,\ell }(n-1) \, p_{E,\ell }(n+1)$, where \begin{equation*} \sum_{n =0}^{\infty} p_{E,\ell }(n) \, q^n := \prod_{n \in S} \left(1 - q^n \right)^{-f_{\ell}(n)},\quad (\ell \in \mathbb{N}, f_1\equiv 1). \end{equation*} We take the product over $1\in S \subset \mathbb{N}$ and denote the complement by $E$, the set of exceptions. In the case of $\ell=1$ and $E$ the multiples of $k$, $p_{E,1}\left( n\right) $ represents the number of $k$-regular partitions. More generally, let $f_{\ell}$ satisfy a certain growth condition. We determine the signs of $\Delta _{E,\ell }(n)$ for $\ell$ large. The signs mainly depend on the occurrence of subsets of $\{2,3,4,5\}$ as a part of the exception set and the residue class of $n$ modulo $ r$, where $r $ depends on $E$. For example, let $2,3 \in S$ and $4$ an exception. Let $n$ be large. Then for almost all $\ell$ we have \begin{equation*} \Delta _{E,\ell }(n) >0 \,\,\, \text{ for } n\equiv 2 \pmod{3}. \end{equation*} If we assume $3,4 \in S$ and $2$ an exception. Let $n$ be large. Then for almost all $\ell$ we have \begin{equation*} \Delta _{E,\ell }(n) < 0 \,\,\, \text{ for } n\equiv 2 \pmod{3}. \end{equation*} Note that this property is independent of the integers $k\in S,k>4$.