Log-Concavity of Infinite Product and Infinite Sum Generating Functions

Abstract

We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let $\{g_d(n){\}}_{d\ge 0,n\ge 1}$ be the double sequences ${\sigma }_d(n)={\sum }_{\ell \mid n}{\ell }^d$ or ${\psi }_d(n)=n^d$. We associate double sequences $\left\{p^{g_d}\left(n\right)\right\}$ and $\left\{q^{g_d}\left(n\right)\right\}$, defined as the coefficients of \begin{eqnarray*} \sum_{n=0}^{\infty} p^{g_{d} }\left( n\right) \, t^{n} & := & \prod_{n=1}^{\infty} \left( 1 - t^{n} \right)^{-\frac{ \sum_{\ell \mid n} μ(\ell) \, g_d(n/\ell) }{n} }, \\ \sum_{n=0}^{\infty} q^{g_{d} }\left( n\right) \, t^{n} & := & \frac{1}{1 - \sum_{n=1}^{\infty} g_d(n) \, t^{n} }. \end{eqnarray*} These coefficients are related to the number of partitions $\mathrm{p}\left(n\right)=p^{{\sigma }_1}\left(n\right)$, plane partitions $pp\left(n\right)=p^{{\sigma }_2}\left(n\right)$ of $n$, and Fibonacci numbers $F_{2n}=q^{{\psi }_1}\left(n\right)$. Let $n\ge 3$ and let $n\equiv 0(\mathrm{mod}3)$. Then the coefficients are log-concave at $n$ for almost all $d$ in the exponential and geometric cases. The coefficients are not log-concave for almost all $d$ in both cases, if $n\equiv 2(\mathrm{mod}3)$. Let $n\equiv 1(\mathrm{mod}3)$. Then the log-concave property flips for almost all $d$.